The words "price" and "cost" are very close in meaning, but there are slight differences. "Price" emphasizes how much the seller has decided to charge for something. "Cost", on the other hand, emphasizes how much you paid for something. So you can say something like this:

The original price was $120, but it only cost me $90 because I had an employee discount.

When a professional like a lawyer, contractor, etc. offers a service, they usually talk about the "cost" rather than the "price". This makes it seem a little more like the amount was not chosen by them personally, but is just a necessary fact.