Short answer: Yes, they will be different, possibly very much so.
Flops are just about floating point operations, so it is a very very crude measure of CPU performance. It is, in general, a decent proxy for performance for scientific computations of certain kinds, but not for general performance.
There are CPUs which are strong in FLOPS - the Alpha is a historical example - but which have more moderate performance in integer computations. This means that an alpha and an x86 CPU with similar FLOPS would have very different MIPS-performance.
The truth is that it is very hard to make a good generic benchmark, though many have tried.