WiMAX and LTE are very similar, so both are "sort of like" Wi-Fi. The key difference is that IEEE standardizes WiMAX, while 3GPP standardizes LTE.
The bigger issue was timing. If WiMAX had gained enough traction to stay at the forefront for another year or two, then we would have had 28-40 nm chipsets, which would have ameliorated many of the early WiMAX smartphone battery life problems.
AJ