From a technical point of view it doesn't appear that there are significant differences between LTE and WiMAX. It's going to boil down to business differences, and the cell carrier backing of LTE seems to be quickly ending the question.
IMHO, this can only matter to those who own stock in WiMAX, either real stock, or let's say professional capital.
Speaking in purely technical terms, there's hardly any difference that could possibly matter to anyone (save the aforementioned). WiMAX does have a much longer pedigree, being part of the same standard that was used to build fixed service MMDS or LMDS networks. In practice, though, you can also build those infrastructures using LTE.
If there's any hope of increasing the competition for broadband service, beyond your local cable company and FiOS, it will come from wireless. But whether that's WiMAX or LTE, who cares?