Does a Universal Turing Machine have more computational power than a non-universal one?
In short yes, a non-universal Turing Machine is not Turing-complete.

The key part is that a computational model is Turing-complete if it can simulate any Turing Machine (or using the transitivity of the simulations, if it can simulate a Universal Turing Machine).

As you note, many Turing Machines are not Universal, they compute something, but they can't simulate any computation that any Turing Machine could do.

A possibly intuitive practical way of thinking about this is that Turing Machines are normally equivalent to programs - they take a particular type of input and compute some output. Universal Turing Machines are equivalent to programming languages or computers - you can build/run any program/Turing Machine on them. Of course I'm using "equivalent" very loosely here.