Conv layers are strictly special cases of FC layers (with respect to expressive power).

> For any CONV layer there is an FC layer that implements the same forward function. The weight matrix would be a large matrix that is mostly zero except for at certain blocks (due to local connectivity) where the weights in many of the blocks are equal (due to parameter sharing).

(per http://cs231n.github.io/convolutional-networks/)

In general, if you can make assumptions about your problem and the form the solution takes, you can find a good solution with fewer parameters and fewer data, at the cost of suboptimal performance on problems that violate your assumptions. Conv layers are an example of this.