Why should I care about lightweight vs. annotated tags?

The big plus of an annotated tag is that you know who created it. Just like with commits, sometimes it's nice to know who did it. If you're a developer and you see that v1.7.4 has been tagged (declared ready) and you're not so sure, who do you talk to? The person whose name is in the annotated tag! (If you live in a distrustful world, this also keeps people from getting away with tagging things they shouldn't.) If you're a consumer, that name is a stamp of authority: that's Junio Hamano saying this version of git is hereby released.

The other metadata can be helpful too - sometimes it's nice to know when that version was released, not just when the final commit was made. And sometimes the message can even be useful. Maybe it helps explain the purpose of that particular tag. Maybe the tag for a release candidate contains a bit of a status/to-do list.

Signing tags is pretty much like signing anything else - it provides one more level of security for the paranoid. Most of us aren't ever going to use it, but if you really want to verify everything before you put that software on your computer, you might want it.

Edit:

As for what to write in a tag annotation, you're right - there's not always much useful to say. For a version number tag, it's implicitly understood that it marks that version, and if you're happy with your changelogs elsewhere, there's no need to put one there. In this case, it's really the tagger and date that are the most important. The only other thing I can think of is some sort of stamp of approval from a test suite. Have a look at git.git's tags: they all just say something like "Git 1.7.3 rc1"; all we really care about is Junio Hamano's name on them.

However, for less obviously named tags, the message could become much more important. I could envision tagging a specific special-purpose version for a single user/client, some important non-version milestone, or (as mentioned above) a release candidate with extra information. The message is then much more useful.

My personal, slightly different view on that topic:

• Annotated tags are those tags meant to be published for other developers, most probably new versions (which should also be signed). Not only to see who tagged and when it was tagged, but also why (usually a changelog).
• Lightweight are more appropriate for private use, that means tagging special commits to be able to find them again. May it be to review them, check them out to test something or whatever.

By default, Git only looks at annotated tags as a baseline for commands like git describe. Think of annotated tags as signposts that have enduring meaning to yourself and others, while lightweight tags are more like bookmarks for your later self to find. Hence, annotated tags are worth using as a reference, while lightweight tags shouldn't be.

Signing a tag is an assurance of the signer's identity. It lets users verify, for example, that the Linux kernel code they've picked up is the same code that Linus Torvalds actually released. The signature can also be an assertion that the signer is vouching for the software's quality and integrity at that commit.