Look them both up on Wikipedia. There is not a tremendous difference between the two and the explanation given so far is pretty whacked out.
Sprint chose Wimax because it was what was available at the time. It is already testing LTE as that seems to be the standard that the industry is settling on. Offering LTE will just require a software change on the base stations.
As far as frequencies, it makes no difference. Wimax is on 2500 simply because that it the frequency range that Clear controls. When Sprint completes its Network Vision program, it will be sporting a multimode network with service on 800-900 (after IDEN is decommissioned), 1900 (CDMA), and 2500 (depending on what happens to the spectrum owned by Clear which Sprint owns 52% of). Higher frequencies can handle more capacity but don't penetrate walls well and don't transmit as far. Low frequencies are just the opposite. Combined, your phone/data device will choose the best range... whether the protocol is Wimax or LTE.
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I work for Sprint, I do not represent them on this forum!