Ok, basic answers here.
Wimax is simply the next generation of wifi for cities.
Wimax was created with one goal and idea, be as much like wifi as you humanly make it, just cover larger areas, lans to mans. (local area networks) to (municipal area networks). Wimax is like wifi, which means once you have it, you have complete control of it. It is mobile broadband.
Companies like intel, google, sprint, comcast, time warner, and a few dozen other created it as an open platform. What I mean by open platform is simple, any device, any network. If you walk past an intel campus, your wimax device will work on their wimax network as well as sprints. If you walk past comcasts wimax network, it will work on that network too.
Regardless of the towers, who own them, which device, wimax device will work. And deeper then that, one of the core ideas behind wimax is you can not block device or service. You can not charge extra for youtube. You can not restrict users to only your websites. You can not filter or reduce in size anything on the internet. As it is given to normal broadband users it was given to wimax users.
The last part of wimax is that in the united states, it was give to the 2.5ghz range. This was done for many reasons, but the number one strength of wimax is the simple fact that it can daisy change together dozens if not hundreds of wimax towers. Which means wimax is created to be more like a web, hundreds of towers all connecting you.
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LTE is simply the next generation of 3g service from cellphone companies.
LTE was created pretty much by Verizon. The main idea was to allow as much control over the network, as humanly possible, to lay with verizon. The LTE group was created to make sure that the people that have 100% control over how the network is used. The big advantages with lte is it can make content available to pretty much anyone. Have a dumb phone, lte can reduce the youtube video to fit that.
But what really helps out is that lte can shape traffic. For example, a customer wants to get directions from California to New York. With the current 3g system, he would probably use google maps. Now there is a lot of people that believe that google maps could get him lost, so with lte, those people could stop google maps services, if he searched for google maps on the market it, it would not appear. If he got the apk from someone, he would be told that he google maps would probably get him lost, so for his safety, he should use verizon's map. It is only $4.99 per month more. Now lte will not allow him to use google maps, lte will just block the service. LTE could offer him the option to use google maps, but that would be $9.99. (not factual, just examples of how lte can make your life safer by controlling what you access on the internet)
You can see how that would help the consumer. I know a lot of people that use their iphone sim, and transfer it to their ipad. LTE will see that the att iphone plan is not the ipad plan, and prevent the user from abusing the network. This would make a lot of people happy because abuse is bad.
LTE, in the usa, is deployed on the 700mhz spectrum. Which means it goes long distances, but gets interfered with by electro magmatic sources. Power lines, cell towers, radio stations, tv's, microwaves, toaters, all mess with 700mhz. Ever turn on your microwave and your tv would get all fuzzy? 700mhz.
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The basic difference between wimax and lte is simply money.
Wimax ideal was created in the same ideal as broadband to your house. You pay a one monthy fee and you can access what ever you want, that is legal. The downside is that small dumb/mobile devices cant really handle the whole internet.
LTE ideal was to make it so verizon and att can offer service to any device they approve. They have 100% control and can charge for any part of the internet they want. Google maps 10 bucks more a month. Youtube 5 bucks. Facebook 5 bucks. Myspace 5 bucks. News sites, 1 dollar per this group of 10 sites. 3 dollars for that group of sites. What to visit facebook, only through our app. Want to visit youtube, only through our app.
The control is very much needed to maximize profits and to off service to every mobile device.