Fatter or larger diameter (higher profile or rim) tires slows handling, this usually leads to a more stable straight line feel and requires more effort to steer.
Narrower or smaller diameter (lower profile or rim) tires quicken handling, making it much easier to change directions quickly, making the bike less stable in a straight line. If the change is radical enough it will make the bike feel twitchy, any movement of the bars or weight shift will cause the bike to move a bit. Some may say the bike has a "nervous" feel.
Wider tires on proper rim widths will possibly brake better, but really that's only if you are braking with the front brake to the point of sliding the front tire. I can brake hard enough with my KLX650 to start to make the 90/90 front tire start to squeal a bit. The Zephyr is borderline, I've braked hard enough I think to start some tread squirm (barely sliding), but I'm not sure and it is braking so hard at that point I'm not sure I could gain much with a wider tire. I'll see how it goes with a set of braided lines next. If I want wider I have to go to a different rim to fit it. The stock rim is too narrow to jump the next size.
Handling as I am speaking of it is the capability to turn in to corners and switch directions (chicane type) quickly. Pretty much that simple.
I've learned this from experience moving both ways in width/diameter on various motorcycles I've owned.
I had a MotoGuzzi and I went for the beefy big tire look - handled like a dump truck. I quickened the handling of a standard GoldWing by dropping one size in width both front and rear - picked it up enormously on winding roads, but not too bad on stability. Finally I was used to the quick handling of my dual sport with the narrow front tire and fairly light weight, that when I got the Zephyr it felt like a truck when it came to cornering and directional change. I resolved it by putting on one profile range lower front tire, from an 80% aspect ratio to a 70% aspect ratio and raised the rear end about 2 inches. Made the bike handle much quicker, which I like. It also made it slightly twitchy at higher speeds, but it doesn't bother me. So there's the practical application of what I've said.
KLX650C, Zephyr 550
SR500, Bultaco Sherpa T