[EDIT]
This answer has been edited. I'm leaving the original answer below for context (otherwise the comments wouldn't make sense).

When this question was originally asked, JSLint was the main linting tool for JavaScript. JSHint was a new fork of JSLint, but had not yet diverged much from the original.

Since then, JSLint has remained pretty much static, while JSHint has changed a great deal - it has thrown away many of JSLint's more antagonistic rules, has added a whole load of new rules, and has generally become more flexible. Also, another tool ESLint is now available, which is even more flexible and has more rule options.

In my original answer, I said that you should not force yourself to stick to JSLint's rules; as long as you understood why it was throwing a warning, you could make a judgement for yourself about whether to change the code to resolve the warning or not.

With the ultra-strict ruleset of JSLint from 2011, this was reasonable advice -- I've seen very few JavaScript codesets that could pass a JSLint test. However with the more pragmatic rules available in today's JSHint and ESLint tools, it is a much more realistic proposition to try to get your code passing through them with zero warnings.

There may still occasionally be cases where a linter will complain about something that you've done intentionally -- for example, you know that you should always use === but just this one time you have a good reason to use ==. But even then, with ESLint you have the option to specify eslint-disable around the line in question so you can still have a passing lint test with zero warnings, with the rest of your code obeying the rule. (just don't do that kind of thing too often!)

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[ORIGINAL ANSWER FOLLOWS]

By all means use JSLint. But don't get hung up on the results and on fixing everything that it warns about. It will help you improve your code, and it will help you find potential bugs, but not everything that JSLint complains about turns out to be a real problem, so don't feel like you have to complete the process with zero warnings.

Pretty much any Javascript code with any significant length or complexity will produce warnings in JSLint, no matter how well written it is. If you don't believe me, try running some popular libraries like JQuery through it.

Some JSLint warnings are more valuable than others: learn which ones to watch out for, and which ones are less important. Every warning should be considered, but don't feel obliged to fix your code to clear any given warning; it's perfectly okay to look at the code and decide you're happy with it; there are times when things that JSlint doesn't like are actually the right thing to do.

TL;DR

Use JSLint if you're looking for a very high standard for yourself or your team, but bear in mind that it's not necessarily the standard, only a standard, some of which comes to us dogmatically from Doug Crockford.

If you want to be a bit more flexible or have some old pros on your team that don't buy into JSLint's opinions or are going back and forth between JavaScript and other C-family languages regularly, try JSHint.

Full version

Two articles with the reasoning behind the fork explain why JSHint exists:

1. JSHint: A Community-Driven Fork Of JSLint

2. Why I forked JSLint to JSHint

The idea behind JSLint is that it's community-driven rather than Crockford-driven. JSHint is generally more lenient (or at least configurable or agnostic) on a few stylistic and minor syntactical opinions that JSLint is a stickler for.

For example, if you think both 1. and 2. below are fine, or if you want to write code with one or more of 1.'s aspects that aren't available in 2., JSHint is for you. If you think 2. is the only correct option, use JSLint. I'm sure there are other differences, but this highlights a few.

1. Passes JSHint out of the box - fails JSLint

   (function() {
     "use strict";
     var x=0, y=2;
     function add(val1, val2){
       return val1 + val2;
     }
     var z;
     for (var i=0; i<2; i++){
       z = add(y, x+i);
     }
   })();
   

2. Passes Both JSHint and JSLint

   (function () {
       "use strict";
       var x = 0, y = 2, i, z;
       function add(val1, val2) {
          return val1 + val2;
       }
       for (i = 0; i < 2; i += 1) {
           z = add(y, x + i);
       }
   }());
   

I find the JSLint code more visually appealing. The only features of it that I disagree with are its hatred of more than one var declaration in a function and of for-loop var i = 0 declarations, and some of the whitespace enforcements for function declarations.

A few of the whitespace things that JSLint enforces are not necessarily bad but are just out of sync with some pretty standard whitespace conventions for other languages in the family (C, Java, Python, etc.) often followed as conventions in Javascript as well. Since I'm writing in various of these languages throughout the day and working with team members who don't like Lint-style whitespace in our code, I find JSHint to be a good balance. It catches legitimate bugs and very badly formed code, but doesn't bark at me like JSLint does (sometimes, in ways I can't disable) for the stylistic opinions or syntactic nitpicks that I don't care for.

A lot of good libraries aren't Lint'able, which to me demonstrates that there's some truth to the idea that some of JSLint is just about pushing one version of "good code" (which is, indeed, good code). But then again, the same libraries (or other good ones) probably aren't Hint'able either, so, touché.

There is an another mature and actively developed "player" on the javascript linting front - ESLint:

ESLint is a tool for identifying and reporting on patterns found in ECMAScript/JavaScript code. In many ways, it is similar to JSLint and JSHint with a few exceptions:

• ESLint uses Esprima for JavaScript parsing.
• ESLint uses an AST to evaluate patterns in code.
• ESLint is completely pluggable, every single rule is a plugin and you can add more at runtime.

What really matters here is that it is extendable via custom plugins/rules. There are already multiple plugins written for different purposes. Among others, there are:

• eslint-plugin-angular (enforces some of the guidelines from John Papa's Angular Style Guide)
• eslint-plugin-jasmine
• eslint-plugin-backbone

And, of course, you can use your build tool of choice to run ESLint:

• grunt-eslint
• gulp-eslint