My comparison for that is the M900 Monster vs. the M750 Monster. Close years, same chassis, same suspension, geometry and setup. (Those early Monsters had non-adjustable suspension, so both bikes had what the factory set up stock.) The 750 has a bit less power, but that isn't the issue at hand. And yes, I could tell the difference, easily between the two. Enough that I have considered swapping the rim to get the 160 on my bike instead of the stock 180, strictly for the handling improvement.If you want to put a 200 on the bike, I say go for it. Just don't plan on doing any hard riding in the twisties or track days. It will definately affect handling in a negative manner.
The dispute about the differences between a 160 and a 180 are hard pressed however. Certainly, if you had a bike with a 160 stock and changed it out for a 180 you would notice a difference in handling. But IMO it's impossible to compare a bikes handling with a 160 to that of a bike with 180. Even if the bikes are similar, the suspension, geometry and setup are different. So it's not really a fair statement to say that you can easily tell the difference between the two.
Overall, I would say that the quickness in steering and turn in between the two sizes is negligible, but it really depends on the bike. Obviously a touring bike from the '80s isn't going to handle as aggressively as a 696, even if they do have the same size tire.