Will a cloned hard drive function exactly as the original?

If I buy a new hard drive and clone my original onto it (using Carbon Copy Cloner or something similar), will my computer function exactly the same as it did before? Will things like licensed programs not work properly? (I can't think of any reason they wouldn't, but you never know.)

Yes. I swap out drives all the time... it's basically transparent on a Mac. I've even booted my MacBook Pro's drive on my wife's MacBook Pro and vice versa. It takes a minute longer to boot as it reconfigures itself but once it starts you can't even tell the difference.

Short answer: Probably

Long answer:

Technically speaking CCC and similiar tools do not make bitwise exact copies of volumes. There are detectable differences (for example, each disk will have different UUIDs). Having said that, I have never seen any software that depends on something like the disk UUID, but it is possible someone uses it as a part of a copy protection scheme or something.

One exception is Time Machine. Time Machine uses the UUID to identify the source disk. So in order for time machine to incrementally continue on your previous backup, you'll have to use the command sudo tmutil associatedisk -a.