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
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.
As has been said, everything should work. It is easy to test this. Simply put the drive in an external enclosure, clone the current drive, then use Startup Disk utility in System Preferences to boot off of the new disk. This will allow you to verify that everything works as intended. If your original disk is mounted, you probably want to eject it to ensure the system is picking anything up off of that disk.
Yes, everything will work the same if you use software such as Carbon Copy Cloner and choose to move the entire hard disk to the new (or "other") one.
It is a great way to have a backup disk that will just work immediately in case if failure of the internal drive (this is how I do with my MacBook Pro), and it's a good companion (not substitute) of Time Machine.