First thing you need to know, THEY ALL USE THE EXACT SAME FOAM!
Sorry for the shouting but I know people are going to argue with me on this and you need to know it.
Now that we have that out of the way, we can get to the real issues. Even though the foam is all the same, there are some differences between tire brands but it's not because one has better foam than another or some such marketing BS.
Wheel design/flex can make a difference. Most have slightly different offsets from each other so going from one brand to another will change your track width which WILL handle differently if you do not adjust for it.
How the doughnuts are cut can make a bit of difference. Some use a small inner hole, forcing the rubber to be stretched over the wheel more. I don't think it makes much difference but I hear it does make some.
Age of the rubber can make a difference. Rubber that was just made 2 weeks ago, mounted one week ago and slapped on the car today will usually have a bit more grip than stuff that's been sitting in a warehouse for a year. Then again, high bite often likes old tires that have been sitting in a hot car for a year.
Durometers vary. One company's 35 can be anothers 37. Many years ago I was running a very unpopular brand and winning regularly with them. Everyone else that tried them said they sucked but they were all trying the 35s (which were really more like 38s) while I was running 30s and 32s. Also the rears were a different offset so if you just slapped a set on your car, your rear track width would be 4mm narrower.
So ultimately there is no hard and fast rules about what is best. You have two options, try everything until you find something that works for you or pick a brand that's affordable and available and work your set-up to match.