Stay with Windows 2003 Enterprise or Windows 2008 Standard?

Becasue I am a nice guy and have genuine intentions to help you,  I am going to say stay on 2003.   We have tried to implement 2008 server in a few areas and so far have had to change it back everytime because of one or more application incompatibility issues.   Only our PDC is able to run 2008 server as of this time and at last heck our backup still isn't real keen on it.  Several important AD / System state backup features don't work.

At least everything works with 2003 server.   If you do update be prepared for the above.  You will encounter several of them on any given change, most can be fixed or gotten around, some can't still.

Are you replacing the physical server with a newer model? If so, and assuming you're going with a new server (or something built in about the last 2 years), I'd go with a 2008 x64 server.

If you're wondering if you should upgrade or not, I'd look at the things 2008 would offer that you may want/use. If this list is small/non-existent, I'd stick with what you have.

Have a number of client on a pure 2008 networks, all were running Win 2000. All programs port over nicely, some with a bit of work. If you have 16 bit programs, a Win 2008 64 bi server is not for you, you would need to run a Win 2008 32 bit server. Be aware Windows 2008 still has SMB signing issues as does the lower OSs, and SMB2.0 may need to be lowered to SMB 1.0 if you go that way.

I have not upgraded any Win 2003 networks over to pure Win 2008, as there is no compelling  reason to do so. Windows 2008 has a few nice additions, but unless you need them the extra work is not justified, especially since you have of 2003 Enterprise. 

Well there you go, with pcmeiners post,  you now have both ends of the spectrum experience wise with the Server 2008 O/S....