No idea why Ducati specs a 170 for the ST2. Consider the only real difference between the ST2 and ST4, which gets a 180, is the engine. A little more HP and I bet that is the only reason there is a different tire spec. As I have mentioned every other time this question comes up, they did the same with my 2001 900SSie. That bike also had 5.5 inch rim. It came with a 170 Michelin Hi Sport. When it was gone, I replaced it with the new at the time Metzeler Sportec M1. The 180 cross-section Metz looked NARROWER than the 170 I had just removed. Why? Different tire profile and design in part, I imagine, but also consider the 170 is probably better for a 5" rim. When you stretch it across a 5.5" rim, you are going to flatten the profile a bit. This could make it "look" wider. Fitting the 180 on a 5.5" rim maintains the tire manufacturer's intended profile and therefore all the R&D and tire review results you likely spent hours pouring over are also maintained...
Handling difference? Who knows. Someone has recently tried a 180 after using a 170 for a long time and claims he can tell a difference thinking the 170 felt lighter. I think there are too many variables to really say for sure. Ride height adjustment, tire pressure, tire design and profile, how you have the bike loaded, how you ride the bike and probably many more. Either way, I think any difference would be fairly minimal and erased from memory after a ride or two unless you're on a track bike trying to shave fractions of a second off your lap times at a particular track...
180s are generally more available as it is a much more common size and it is the right tire for a 5.5" rim. You can't go wrong with one. You could also use a 170 and live a happy life as well if that is what you choose!
I wanted to add, but got side tracked...
Keep in mind there is more to it than "180 vs. 170". There is also sidewall height. The 180 comes as a 55 and the 170 comes as a 60. This is the ratio of sidewall height to width. 170/60 would have a sidewall height of 102MM and the 180 would have a sidewall height of 99MM. This is where tire manufactures can differ. Where and how do they measure sidewall height? In general, the 170/60 will have a larger diameter than the 180/55. This may have as much of an impact on a said lighter feeling than the width of the tire as it would be similar to raising ride height a little. Less of an effect if it is stretched across a too large rim, but none the less it is there. I still think it's pretty minimal, but for academic purposes, it needs to considered in the discussion.
I used this link toplay with three tire size configs.
http://www.gearingcommander.com/ You can see the results in the pic.