Can anyone explain the difference between the two?
x86 denotes 32-bit machine architecture.
amd64 denotes 64-bit machine architecture.
Which version is more reliable for gaming?
Depends.
x86 is reliable for 32-bit games.
amd64 is reliable for 64-bit games.
Postscript:
amd64 is commonly referred to as x64. The previously used 64-bit version was designed by the Intel corp which was called as x86-64. It was not compatible with the native x86 instruction set. In 2003, AMD introduced the perfect 64-bit version for the x86 instruction set. Since they introduced it, it is named after them as amd64. Hope you would have enjoyed the history :)