The advantages of the 700x25 you're now riding is that they will absorb more road shock. They will also give you a slightly larger contact patch on the road for braking and cornering.

The advantages of the 700x23 are lower rolling resistance (i.e. they're faster). No reason you can't ride these at your weight. Tire pressure is critical. You'll increase your chances of flats (especially the pinch variety) if tire pressure is too low. You want about 110-120/rear and 105/110 front. If you haven't ridden the bike in a few days, you'll need to "top off" the pressure since your tires will have lost 5-10 lbs.
The only other thing you may need to invest in is different tubes if the ones you're running are too wide to install comfortably in the narrower 700x23 tire.

The reason you're finding more selection in 700X23 is that this is the most popular size. If however, you value comfort over speed, I'd stick with the 700x25.