How important is it to have 64-bit apps on a 64-bit OS

Now that more and more computers are shipping with 64-bit processors and 64-bit operating systems, just how important is it to have 64-bit applications?  What are the benefits of 64-bit applications?

I've been using an HP dv6661se laptop with an AMD 64-bit processor for four years.  In the early days, there were quite a few frustrations such as finding hardware drivers and application compatibility issues.  (There was one particular frustration with a lack of 64-bit drivers for my USB-to-serial adapter.)  Today, the compatibility issues are much less of a challenge than they used to be, but what are some of the issues and benefits?

Adobe Premiere CS5 only runs on a 64-bit system.  I'm sure there are other apps out there that will only run in a 64-bit environment.  This is almost exactly like the move from the 16-bit to the 32-bit world.

I ran across this discussion at superuser.com that may be helpful.