If the equivalence principle is true, and gravity causes time dilation, why doesn't acceleration also cause time dilation?

I think the confusion here stems from trying to learn a concept by an accumulation of isolated statements in words. That may be possible for some concepts but it is not the right approach with something like time dilation. Rather, you come to understand it by building up a grasp of the wider subject of spacetime, first as it is treated by special relativity, and then by general relativity. Asking about the impact of acceleration on time dilation is like asking about the impact of acceleration on a total journey time in Newtonian physics: it is a relevant part of the physics but it does not on its own tell you very much (e.g. a large acceleration does not necessarily make the journey time smaller; it might make it longer if it is in the wrong direction). Time dilation is the name we give to the fact that the accumulation of proper time along certain worldlines is different from the accumulation of proper time along certain other worldlines, and it is connected to the way events in spacetime may or may not be said to be simultaneous depending on how systems of coordinates are set up. In the twin paradox what acceleration causes is the departure of one of the worldlines from a line of most proper time.

Here is the twin paradox in a nutshell.

Suppose there are two events $A$ and $B$ with a time-like separation. These events could be, for example, the departure and return of a rocket on planet Earth. Of all the timelike curves extending between these two events, there is a unique one along which the accumulated proper time is the most. That unique line is the one followed by a body undergoing inertial motion. For all other lines the accumulated proper time is smaller.

That's it. You can now add some story colouring by having one twin follow the unique line of most proper time, and another twin follow some other worldline. This helps to make clear that by "proper time" we mean the amount by which ordinary physical things like wrist-watches, hearts and bodies proceed through their ordinary physical evolution in their own instantaneous rest frame at each moment.

The Equivalence Principle states that being in an accelerating reference frame is indistinguishable from standing still in a gravitational field. We know that gravity causes time dilation, so why not acceleration?

1: We know gravity causes time dilation

2: We know acceleration does not cause time dilation

3: We can distinguish between gravity and acceleration

If we kidnap a guy, put him in small windowless room, said guy can not tell if he is in an accelerating rocket or not.

The Twin Paradox is supposed to be resolved by the acceleration effect. Otherwise the time dilation is completely reciprocal and no difference will be seen between the clocks. Therefore acceleration must be causing time dilation?

Well, it is said really often that acceleration is important. It is said by amateurs mostly. Also acceleration not being important is said quite often, as some wiser folks try to correct the amateurs.