SSE is a superset of x86. AMD64 is a different binary architecture. That is why there is a different binary stream for it in the Chess.app. That is why a program has to choose to run in AMD64 mode or x86 mode, not both simultaneously. That is why 64-bit QuickTime has to do IPC to load 32-bit codecs. That is why 64-bit Safari has to use IPC to load 32-bit Flash. The only truth to Wikipedia is the marketspeak: AMD64 chips boot up in x86 mode, and with OS support, can have processes executing in AMD64 or x86 mode. The kernel can run in x86 mode, while an app runs in AMD64 mode. (Hence Leopard) But one process can't do both, because AMD64 is NOT a superset of x86.
Is AMD64 binary compatible with x86? Very close, but no. The INC and DEC instructions are taken away, replaced with the REX prefix that controls access to 64-bit and the new registers. New binary machine code is used for INC and DEC instructions. This precludes (along with the whole 64-bit address thing) AMD64 from being a superset of x86. AMD kept AMD64 as close to x86 as possible, so that minimal reworking of existing compilers, assemblers, and assembly code would be needed to transition to AMD64. They were quite successful, obviously, with FreeBSD available on AMD64 in June 2003, and Windows XP in 2005. (Microsoft had to wait for Intel to ship their 64-bit processors, or else they would have seriously embarrassed their good buddy.)
If AMD64 was constrained by x86 (to be a true superset), it wouldn't have been able to do all the things that make it such an improvement. You think Intel wouldn't have already done those things 20 years ago, during the RISC vs. CISC battle, if it was that easy? No, the best they could do was SSE, which added some registers, but also imposed rules on how SSE instructions would execute and intermix with the non-SSE instructions. In addition, the OS had to be modified to make sure the new registers would be protected during context switches (aka multitasking). Windows 98 was the first version that allowed SSE to work. SSE pushed things the farthest you can go before you have a new architecture.
If you want technical data, try developer.amd.com.