Well, first we must make the distinction that just having a larger OD coil doesn't mean it has better power handling. If we assume the same winding height, number of windings, coil material, and convection cooling, a 4" coil will, indeed, handle more power. Otherwise, it may be a different story. It's a "sum of the parts" kinda thing.
Keeping with that same assumption....a 4" coil will have more moving mass. It will have more coil in the gap, increasing BL. Although slightly offset by the increase in mass, this results in higher efficiency. Although a larger diameter coil generally has lower inductance, a 4" coil with the same winding height will actually have higher inductance than an equivalent 3" coil because the winding length is higher. This generally means earlier rolloff on upper frequency response.
As mentioned above, the implementation of a larger OD coil has a challenge when integrating with the spider. A wider diameter coil means that, in a given basket, there is less spider area. This means that your stroke, particularly linear stroke, can be limited. You can minimize this by increasing the height of the corrugations in the spider, but you eventually need a wider diameter spider, meaning wider basket, meaning custom tooling, meaning $$$$. With large displacement systems, the suspension is almost always the limiting factor.