WiMax and LTE were designed to move the existing scheme of separate cellular and data infrastructures over to one more simplified data technology or "All IP Network" (AIPN) which means you would only need one radio for voice and data in a phone which helps reduce size and power consumption. But conceptually you have to understand that cellular companies are actually getting out of the cellular business in favor of becoming a mobile broadband service provider.
Initially they were not happy about having to do this (change is scary) but they saw that if they didn't they would then loose out to other companies who would. At this point they now see the benefits including lower overhead to operate and manage the network, increased bandwidth, capacity, increased security and simplified worldwide roaming. This means that there is a lower cost per Bit due to spectral efficiency and flexibility with WiMax and LTE.
They also see opportunities for new revenue streams that currently would be difficult or impossible. Think blending voice with other services: such as presence, messaging and social networking. Something like handing off a phone call to a laptop or tablet could be an easy carrier based service. On demand HD video and HD voice are also things likely to come.
All this of course is not going to happen overnight and it will be years down the road when transition is complete but shorter for early adopters who are willing to switch over to a capable phone once the service is built out.
Hope this helps.