WiMax is an LTE alternative designed by an IEEE 802 committee. It is designed for last mile delivery and also meets the requirements to be considered a 4G and 5G protocol.
Although technically superior to LTE, phone companies (outside of Sprint, in the US at least) chose not to deploy it as it was not based on existing protocols (GSM and CDMA), even though that isn't an actual legitimate reason, as "true" LTE requires VoLTE deployment, which phone companies refused to do until forced to; VoLTE-enabled handsets perform a lot better than their pre-VoLTE or VoLTE-disabled siblings purely because of not needing to waste valuable spectrum on 3G connections, allowing that band to be reassigned to a 4G radio on the tower.
On top of the VoLTE debacle, companies invested in the GSM and CDMA monopoly tried to claim WiMax did not perform well at long distances, the same distances that LTE does not work well with today (found frequently in rural areas in the US, or similarly in areas heavily shadowed by hills or tall buildings); however the adoption of 600/700mhz to fill in those gaps (plus the forced adoption of VoLTE to improve spectrum usage) has proven that to be false.
LTE in areas that are partly covered by existing bands, with the gaps filled in by 600/700mhz, have finally caught up to WiMax in real world testing.
Interestingly, Asia has adopted WiMax heavily but may be switching to LTE in the future for 5G deployment, even though WiMax beat LTE-A to the commercial gigabit deployment milestone, due to these continued misconceptions. Africa's few networks that were WiMAX are (or have already) switched to LTE (driving up the cost, and lowering the reliability of their networks).
The one place WiMax survived in the US was fixed broadband links (as this is what WiMax was originally designed for, until it merged with Korea's WiBro spec more than a decade ago), but that seems to be finally being replaced in favor with LTE-A's fixed profiles.
WiMax will probably beat LTE to working gigabit deployments, although you'll need to live in Asia for this to be relevant to you.
WiMax is not related to WiFi; although the actual underlying technology of all standards are rapidly converging, they are designed for different purposes.