Before anyone gets angry lets start by saying both methods can produce barrels that are beyond human capacity to shoot.
Button barrels are made by swaging the grooves, either pulling or pushing a carbide or other very hard material through the barrel and displacing the metal and not cutting it. This induces a lot of stress in the barrel. This also must be done while the barrel is in full blank size to prevent a tapered hole. The barrels are stress relieved to get them back to "normal". I think most of the time they are contoured at this stage. They are then lapped to get a consistant bore diameter.
Cut barrels are rifled after they are contoured then the grooves are cut at about .0001 per pass to desired depth, then lapped.
The differences in my OPINION... The twist is probably more consistant across the cut rifled barrel. In a cut barrel the twist rate is conrolled by turning the cutter as it passes down the barrel. Button barrels the twist is mostly controlled by the shape of the button and the twist rate can vary depending on many variables. The proof to me that cut rifling is more consistant than button rifling is the fact that quite a few button barrel makers offer different grades of barrels. I have never seen this from the cut barrel makers. The cut barrels can also be ordered with more custom twist rates, groove profiles and diameters. It is much easier to change the gear ration that creates the twist rate on a cut machine than get a button made specifically for one barrel. The down side to cut rifling... it takes much longer to rifle the barrel, several hours vs seconds...for the rifling.
It's really about personal preference since both can make a barrel that shoots better than we do. I see advantages to the cut process and that makes me a little happier when I pull the trigger so that's what I shoot....mostly....LOL