One of the reasons that the 4E characters feel a bit the same to some people is that the scope of what each character can do is rather narrow, mechanically. So this in turn means that if start varying the characters by taking away what some can do, you'll make them different from other classes, but more alike within that class. This is the early D&D fighter--picks a weapon, hits hard, takes a beating, color with a name and some background.
Or if the problem at the meal is that the menu is rather limited, you aren't ultimately helping by giving all the peas to Susy, the carrots to Bill, the pot roast to Mark, and the gravy to Jane.
Instead, I'd look to first expand the options that are available to all classes. That is, you need a lot of mechanical ways for them to act, decision points, etc. And if you ultimately want the classes varied mechanically, make more options than you strictly need. Have 10, and give each class 7 or 8 of them.
With some serious thought, you could probably design these options to cover a lot more fictional space, and thus the connection to the mechanics would be more natural.
For example, accept that there is only so much you can do with knocking off hit points in different flavors, with each character having a standard attack (ignore move and minor for a moment). But if you go with the 2 action per turn (but not two attacks) idea that I suggested elsewhere, then you can have some options take both actions.
Maybe arcane dailies take both action, have some extra power, but penalize the caster in that being all they can do that round. If they cast an at-will, lower powered spell, they can still move or drink a potion or whatever. The standard/move/minor economy doesn't really support this kind of change (minor sustain not withstanding). So if you want to meaningfully vary power by the action economy, change the action economy to have enough granularity to vary it. Fighters, OTOH, don't get dailes. They get stances, which can be used over and over, do nothing in themselves, but modify other attacks, and take an action to activate. Again, doesn't work so well in the current action economy, but does if the stance switch has a cost but still allows an attack.
I think it is obvious how those mechanics relate more closely to what people expect in the fiction. But you'll note that this only works in a balanced way
and connects to the fiction if you build in enough levers to have those kind of trades. Otherwise, things have to get watered down to where they no longer match the fiction, or be overpowered.
This is really the same design problem as the 3E wizard versus prestige classes. When all the wizard has to trade in for a prestige class is the familiar, the prestige class can hardly be worth taking. (What it does in the fiction is not supported by its mechanics.) But if the wizard loses caster levels, it is too much. He isn't much of a wizard anymore. So we had six dozen attempts to work around this intractable problem instead of giving all the classes more to start with, so that prestige trades would make sense.