The extra registers affect the way programs are written. You get a more RISC-like feel: this shows up most notably in the calling conventions, which pass all arguments in registers most of the time. You see more RISC-like three register arithmetic because the compiler has more opportunities to use LEA. Etc.

It is a lot more than that. Calling conventions, register names and usage, etc. etc. I have a "cheatsheet" sitting on my desk that I wrote up last time I need to do some x86-64 reversing. Do no sit down with a x86-64 binary and use this manual as a guide, you will not get very far.