Are exploits platform dependent?
As in, does that mean that the IE vulnerability needs to be developed IN ASSEMBLY once for each OS( XP, Vista, 7, 8 ), doubled based on 32/64 bit, then doubled again based on byte order?
The use of assembly doesn't make it more platform dependent. It's only just as platform dependent as it would be if written in say, C, (although for most purposes C makes it much easier to target those different platforms).
The page you linked doesn't indicate which platforms the exploit targets, and only shows x86 code. If it also targets x64, that's still only two platforms: x86 Windows and x64 Windows, and even for those two the code would be largely isomorphic (similar). The mere fact that it has assembly in it doesn't suddenly explode the complexity of supporting different editions of Windows which run on the same CPU architecture.
(Just to hammer the point: portable assembly can be written and compiled once, such that the same blob of machine code will then execute under all of Windows, OS X, Linux, and no operating system at all, on the same CPU architecture. In this way, assembly is less platform-dependent than other languages, because it requires less of a platform. The standard executable file formats that house machine code are still OS-specific, though; for example OS X won't know how to load a Windows .exe, even if the machine code inside could run if it were loaded.)
Byte order is not an issue as x86 & x64 are purely little endian.
Also note that the assembly you're seeing there is a disassembly, so it was not necessarily written in assembly; it could have been written in nearly any language. As this is an exploit, and designed to do weird extralinguistic things, the hackish parts of it would have been easiest to write in assembly itself, and some may have been easiest written in C, and some may have been started in C, disassembled, and adapted into the assembly.
So the short answer is, supporting different Windows versions in the exploit is not as difficult as you're imagining it to be.