There is a good article on GigaOm drawing comparisons between this two technologies. My comment :
I think LTE being based on a flat architecture, there are more parallels between WiMaX and LTE than the differences. In long term, the evolution (Pun intended) would be driven by cost factor and timing.
WiMaX at this stage has benefit of timing, whereas LTE trumps from cost factor (upgrade path). If 80% operators move towards LTE in the future, then it remains to be seen how much interoperability would be implemented between this two competing standards and how much the big operators would be willing to make that happen.
Having said that, LTE has interim solutions as mentioned by Clonmore, whereas WiMaX is working on 802.16m to match LTE on performance levels. Looking at whole picture, we see quite similar architectures in 2012, but question is how do you get there ?
Disclaimer: I work for NSN and have worked on both technologies and do not advocate any one in particular.