I can bank on the Patriots (an American football team) winning this year. I can count on you to do this task for me. I can always rely on you to do something for me. The difference between each usually depends on the context. But to "bank on" means I can 'bet on' something and expect that I will win or be right on my guess or expectation. To count or rely are interchangeable and they mean you can depend on something, or you can depend on someone to complete a task or favor you, or to even rescue you from a negative situation.

Sometimes the use of "bank on" involves money. For example, someone might be "banking on" a business deal doing well because they have invested money in it. The use of "rely on" might involve cases of need and dependence. For example, someone might rely on their friend to give them a ride to work because their own car is not working. The use of "count on" might involve cases of someone doing a job that they are expected to do. For example, the soccer coach might count on his star player to get a goal. Hope this helps