I know that x87 has higher internal precision, which is probably the biggest difference that people see between it and SSE operations. But I have to wonder, is there any other
There is considerable legacy and small system compatibility with the x87: SSE is a relatively new processor feature. If your code is to run on an embedded microcontroller, there's a good chance it won't support SSE instructions.
Even systems which don't have an FPU installed will often provide 80x87 emulators which will make the code run transparently (more or less). I don't know of any SSE emulators—certainly one of my systems doesn't have any, so the newest Adobe Photoshop elements versions refuse to run.
The 80x87 instructions have good parallel operation characteristics which have been thoroughly explored and analyzed since its introduction in 1982 or so. Various clones of the x86 might stall on an SSE instructions.