What is the difference between a check and a paycheck?

There is little difference. A paycheck is a type of check used to pay wages. These days many people opt for direct deposit. So, the term paycheck can also refer to the payment itself:

1: a check in payment of wages or salary

2: wages, salary

http://www.merriam-webster.com/dictionary/paycheck

A paycheck is simply a check for your salary. It's just like a rent check, or a birthday check, or a grocery check... I've had "paychecks" that were personal checks from the owner of the business, I've had ones that are printed in the office I worked in and signed right there, and I've had paychecks that are printed through a third party company and mailed to me (my favorite, of course, is to forgo the "paycheck" entirely and get direct deposit :) ). Really, they're all just checks.

Although that's a little disingenuous, because banks are often slightly more trusting of paychecks. However, this has little to do with it being a "paycheck," per se, and more to do with the fact that they see you getting the same check for (roughly) the same amount on a regular basis; having seen you get a paycheck for the same amount from the same company for the last 12 months, there is less risk of the check bouncing or being returned unpaid, so you can often get banks to waive their hold policy and just give you the money.

I think the only difference would be that someone writing you a personal check could always cancel the check for whatever reason they choose and not have much in the way of legal consequences (unless they used it to buy something from you and are now trying to get out of it without returning the item(s)).

With a paycheck, the issuer is acknowledging that they owe you for services (in the form of labor) rendered, so it's an implied "contract" of sorts for which they'd have to have VERY good reasons for trying to cancel once they've given it to you. In addition, they not only issued the check to you, but they separately paid taxes on the wages, so i creates a bit of an entanglement if they try to cancel the check without good cause.

In either case, it's still just a check, and it's good for about six months from date of issue.