Why should I pay for an SSL certificate?

For most uses, there's no good reason to pay for them.
See the very bottom for a summary of the exceptions.

Let's take a step back and explain what certificates do and roughly how.

What are commonly called "certificates" consist of two linked pieces:

The certificate proper, which contains a public key and some identification (such as a domain name).
The private key, which allows the holder (and only the holder) to digitally sign messages in such a way that they can be verified using the above certificate.

If you want a certificate for yourdomain.com, you:

Create a private/public keypair, and keep the private part, well, private.
Ask a trustworthy third party ("CrediCorp") to create a certificate for yourdomain.com with your public key.
Prove in some way to CrediCorp that you control yourdomain.com.
Put the private key and the obtained certificate on your server, and configure the web server to use them.

Then, if Alice visits yourdomain.com, her browser gets the certificate from your web server, along with a message signed by your private key. Then her browser checks three things:

That the signed message can be verified by your certificate (proving it's been signed by the corresponding private key that only yourdomain.com should have).
That the certificate's domain is the domain the browser is trying to visit (yourdomain.com).
That the certificate is from CrediCorp.

The combination of these three things guarantees Alice that she actually is talking to yourdomain.com, and not to some impostor... Provided that Alice trusts CrediCorp.

(There also follows some crypto voodoo dance to turn this authenticity into confidentiality.)

How does Alice trust CrediCorp?

That's the real crux here. In short, at some point CrediCorp said "Hey, we're going to make certificates". After putting in a lot of effort following a lot of rules, they managed to convince some people that CrediCorp are, indeed, trustworthy, and they will only issue certificates correctly.

In particular, they managed to convince the makers of, say, Firefox. As a result, CrediCorp gets on Firefox' A-list, and their certificates are trusted by Firefox by default. So really, Alice trusts Firefox, Firefox trusts CrediCorp, and CrediCorp trusted (after verifying) you when you claimed you controlled yourdomain.com. It's almost like a chain.

But, Firefox doesn't just trust CrediCorp to issue certificates for yourdomain.com, it trusts CrediCorp certificates for any domain. And Firefox also trusts ShabbyCorp, for any domain.

This has consequences. If someone manages to convince ShabbyCorp that they control yourdomain.com (because ShabbyCorp isn't very thorough), then they can obtain a ShabbyCorp certificate for yourdomain.com with corresponding private key. And with that certificate they could impersonate your web server. After all, they have a certificate (plus key) for yourdomain.com that is trusted by Firefox!

CrediCorp and ShabbyCorp are what's called Certificate Authorities, CAs for short. In the real world, ComodoSSL and Let's Encrypt are examples of CAs. But there's a lot more of them; as of this writing, Firefox trusts 154 CAs.

Whoa. But how does that answer my question?

I'm ehm, getting to that...

Here's the thing. The mechanics I outlined above apply to all certificates. If you have a correct, trusted certificate for your website, it'll work. There isn't anything special about Brand A certificates versus Brand B certificates; they're all subject to the same CA requirements, and the same crypto math.

And even if you like CrediCorp better — because you know, they just sound sooo much more trustworthy — using them won't really help you. If an attacker can convince ShabbyCorp to give them a certificate for your site, the attacker can use that certificate to impersonate your site, regardless of where you got yours.

As long as Firefox trusts ShabbyCorp, visitors won't see the difference. (Yes, visitors could pull up the certificate, and dig through there, see who issued it. But who does that?) As far as forging certificates is concerned, this makes the entire system as weak as the weakest of 150+ CAs. Why yes, that is scary, and it's probably the biggest criticism people have of this entire scheme. Still, it's what we're stuck with.

Point is, if you don't trust a CA to give out "good" certificates, getting your certificates elsewhere doesn't help you much.

Gotcha, everything is equally doomed. No caveats?

Weeeelllll...

Let's start with killing the point I made in the last section. Nowadays it is possible to lock your domain to just CAs of your choosing using DNS-CAA. Suppose that you do trust Comodo, and don't trust other CAs, it is possible to request all CAs other than Comodo to not issue certificates for your domain. In theory. (Because DNS-CAA is not checked by browsers, only by issuing CAs. So a compromised CA could ignore this safeguard.)

If you're willing to go through that trouble, then the question becomes: is Let's Encrypt actually less trustworthy? Or less secure? Trustworthiness is a hard one, how do you quantify that? All I can say is that in my perception Let's Encrypt is no less trustworthy than other CAs. As for the security of their validations, they are very similar to what commercial CAs do (for DV certificates anyway). See also this question.

For what it's worth: the StackExchange network, which this site is a part of, currently uses Let's Encrypt certificates. Most people would never notice this, and if they did I sincerely doubt if it would mean much to them.
For a certificate to be meaningful, the issuing CA must be trusted by software vendors, otherwise the certificate is useless. I used Firefox as an example, but really you want the CA to be trusted by at least current and somewhat older versions of Firefox, Chrome, Windows, Mac OS X, iOS, and Android. And the dozens of smaller players. CAs worth considering (that includes ComodoSSL and Let's Encrypt) are trusted by all these entities.
If a CA misbehaves, or is revealed as untrustworthy, it will get removed from the various trust stores quickly enough to ruin the day of certificate owners. Two notable examples I know of are DigiNotar and StartCom/WoSign (check out the articles, they provide interesting insights in the trust dynamics!). So if you think Let's Encrypt will screw up, or will be dropped for some other reason, not using them will prevent you from getting caught in that particular fall-out.
Certificates employ some crypto math magic; the question is which crypto math magic? What if it's weak magic? This is actually a real concern, and CAs have shown to drag their feet on upgrading this, too. Luckily, browser vendors have picked up the slack by setting minimums here for certificates to be accepted. For example, certificates using RSA-1024 or SHA-1 are now rejected by most browsers, so any certificate that works in practice doesn't use these deprecated crypto primitives. The upshot is, it's pretty hard for any CA (Let's Encrypt included) to disappoint on this part anymore.
Before, I more or less said that all certificates are created equal. I lied, they're not. In particular, what I discussed up until now are "Domain Validated (DV) certificates", which are what the vast majority of websites use. They provide a measure of certainty that your browser is actually talking to the domain it shows in the URL bar. There are also "Organization Validated (OV)" and "Extended Validation (EV)" certificates, which require much more elaborate checks from CAs. In particular, you should only be able to get an EV certificate for somebank.com / SomeBank Inc., if you can actually prove you are SomeBank, Inc.

EV certificates are a lot more costly to obtain (ballpark: hundreds of EUR/USD per year), and they may be rewarded with a green URL bar or padlock in the browser, maybe displaying "SomeBank, Inc." as well. Contrary to DV certificates, they also offer some idea as to who the website might actually belong to. The upside is, they may look more legit. The disappointment is, users rarely pay attention to them, so their effectiveness is limited.

An attacker with a forged DV certificate can still impersonate the site, just without the extra visual clue an EV certificate may offer, and users generally don't notice the distinction. Conversely, it is possible to obtain a misleading EV certificate to make phishing easier. As a result, both Chrome and Firefox will be dropping their visual nods to EV certificates, and some people believe they will go away entirely.

If you're a bank, you probably still want an EV certificate for now. Otherwise, not so much. But if you do need EV, Let's Encrypt isn't for you because they simply don't offer EV certificates.
Certificates are only valid for a limited time. Certificates from a typical commercial CA tend to be valid for one year, but I've seen anything from three months to three years. Let's Encrypt certificates are valid for 90 days, which is on the short side of that range, so you'll need to renew them often. For Let's Encrypt users, this is usually automated so that certificates are replaced every 60 days.

Being able to automate renewal with widely available software is actually more pleasant than the yearly Oh shit my certificate expired? What's my login at the CA? How does this work again? ritual that most small sites seem to end up with at commercial CAs.
Before, I called it scary that there are so many CAs we all have to trust. Having many CAs is also an advantage though, in the sense that removing a single one from our trust stores has a limited impact on users. In particular, expelling a single CA will only affect the certificates issued by that one CA. If everyone ends up using one single CA (which some people fear might happen with Let's Encrypt), we concentrate all of our trust there, and lose the advantages of that fragmentation.
And finally, there's other benefits a paid CA might offer, such as commercial support, or a million-dollar SSL warranty. I have little faith in both of these aspects, but they are things that Let's Encrypt does not offer.

My head hurts... I had a question, I think?

Use what you feel comfortable with! For DV certificates, there is little that actually differentiates the various CAs. I use Let's Encrypt both professionally and privately, and I'm happy with it.

There really are only four potential reasons I see to avoid Let's Encrypt:

If you need EV (or OV) certificates.
If you can't or don't want to automate certificate renewal and three months certificate validity is too short for you.
If you don't trust Let's Encrypt (but be sure to consider other measures like DNS-CAA as well, and you should probably blacklist Let's Encrypt in your browser then, too).
If you believe Let's Encrypt will be discontinued or dropped from browsers for some reason.

If none of those apply to you, feel free to not pay for your certificates.

Let's Encrypt is superior in many ways, including the ones that you have mentioned, such as:

It's free. Hard to get past that.
It has automatic renewal (I'm sure it's not JUST exclusive with Let's Encrypt, however)
It's pretty easy to set up.
Google and many others support it as a trusted CA, which is a huge deal when it comes to SEO and security.

However, there are a couple of cons.

The verification system that it works on to make sure that, you, well, own the site, is not compatible with some website hosts, I have had a fair amount of headache trying to get Let's Encrypt work on InfinityFree and I just accepted the fate that I couldn't do it.
You don't get any kind of insurance that says "If this breaks, we'll help you out" since it's open-source, you are on your own if Let's Encrypt doesn't work or is somehow cracked.

LetsEncrypt certificates are great. I use them myself instead of buying certificates. There are a few drawbacks:

LetsEncrypt certificates only last 3 months. Most purchased certificates are good for one or two years. That means that you absolutely need an automated process in place to renew your certificates or it is going to be too easy to forget.
LetsEncrypt only offer the lowest validation type of certificate. Domain Validation (DV) only validates that the owner of the certificate has control over the domain. Organization Validation (OV) certificates also check the documentation of the person or company requesting the certificate. Extended Validation (EV) certificates require even further checks. The better your certificate, the harder it is to be forged, and the more your site's authenticity can be trusted because of it. In practice, browsers only give a visual nod to EV certs, usually showing something in green in the address bar for them. Up to this point, most users don't know or care about the different validation levels.
Wild card certificates are a bit harder to obtain from LetsEncrypt. From other places you generally just pay more money. LetsEncrypt requires DNS validation for wildcard certificates.

Historically, security certificates have always cost something. Other companies that have offered free certificates have come and gone. I used to use StartSSL which offered a single domain free certificate until they did some shady stuff and browsers stopped trusting their certificates. LetsEncrypt has fewer limits than previous free certificate vendors and is far more automated. It also has some big supporters such as the EFF, Mozilla, Chrome, and Cisco. See https://letsencrypt.org/sponsors/ It appears to be well enough run that I expect it to be around for years.

Why should I pay for an SSL certificate?