What is the difference between grills when frying a burger?
There are many styles of burger, so there is no single answer.  Steamed sliders are going to have different constraints than a smashed burger compared to a pub style burger.

For burgers that are griddled (for the purposes of this post, cooked on a hot, flat surface), the main issue is how much heat can get into the burger, and how fast; secondarily, there are issues of sticking and general maintainability.

Some hamburger cooking methods rely on sustainable, very high heat to sear the outside of the burger, and give a good crust, even while maintaining a less cooked intererior.   This is dependent mostly on the total thermal mass of the pan.


Stone - Lots of mass, but low thermal mass.   Doesn't transmit heat very fast.  Also requires considerable care and maintenance to prevent sticking.
Pan - There are so many types, it is hard to give a single characterization.  Thicker, heavier metal tends to be better.   See Navigating the different types of cookware - new kitchen advice for some general advice.
Cast Iron - Generally has high weight, and very good thermal conductivity (compared to any non-metal), so often ideal for high temperature searing of hamburgers.  Requires seasoning and maintenance.
Charcoal (or propane) grill -- can develop extremely hot radiant heat, which is ideal for some forms of burger cookery, but cannot be done inside.


However, there are so many different types of tools and styles of burger, the quality of outcome is going to depend on your skill, what you are trying to achieve, and how you use the equipment you have as much as the specific pan that you use.

Similarly, the flavor is going to depend, not only on how you cook (which can develop the seared, brown, meaty flavors), but how you season the meat, and the quality of meat you start with.   This is again as important as the particularly pan that you might use.