Use jQuery or Q.Js for promises

Both are based on the Promises/A standard and implement a then method (though only current jQuery, they once had a incompatible pipe instead of then). However, there are a few differences:

• Q has exception handling. All thrown errors in the async then callbacks will be caught and reject the promise (and will only get re-thrown if you call .end()). Not sure whether I personally like that. It's the standardized way which jQuery does not follow, rejecting from then in jQuery deferreds is much more complicated.
• Q promises are resolved with a single value/reason (like you return/throw it from then), while jQuery allows multiple arguments in resolve/reject calls on its Deferreds.
• Q has lots of Proxy methods which will allow you to modifiy future values
• Q has .all and similiar, which are more complicated with jQuery ($.when.apply($, […])).
• Q does explicitly work with ticks in the event loop and guarantees asynchronity, while jQuery can be synchronous as well. This is now required by the Promises A/+ specification.

… which is basically Promises/B. As you can see, the Q API is more powerful, and (imho) better designed. Depending on what you want to do, Q could be the better choice, but maybe jQuery (especially if already included) is enough.

JQuery's promise implementation of the Promises/A spec has some real issues. The following link describes them far better than I can: missing-the-point-of-promises

Bergi's answer covers things fairly well. I wanted to add, though, that we've created a guide for Q users coming from jQuery. To summarize the relevant sections:

• Q handles exceptions, allowing you to handle all errors through a uniform interface.
• Q focuses on chaining with all its methods, whereas jQuery only allows chaining from then/pipe.
• Q promises guarantee asynchronicity, thus avoiding the control flow hazards and race conditions that result from jQuery's sometimes-sync, sometimes-async behavior.
• Q promises are always fulfilled with a single value or rejected with a single reason, just like synchronous functions always either return a single value or throw a single exception.
• Q enforces a separation between the deferred and the promise, whereas jQuery merges them into one object with the option of separating them.
• Q does not track a context object along with the fulfillment or rejection, since this has no parallel for synchronous functions (i.e., you never return a value as well as a this in which the caller must run). So there is no resolveWith or rejectWith.
• Q uses Promises/A+ terminology; the main difference is that Q uses "fulfilled" where jQuery uses "resolved," and in Q "resolved" means something more subtle.

The guide also contains a table paralleling jQuery and Q promise APIs.