Btw, cyclocross bikes have a higher bottom bracket, so they have more clearance for off road... (typically). They have burlier wheels, etc. to deal with it as well. Typically on a cyclocross bike you want more standover height than you want on a road bike, just because of how you use it... So you'd want to get a smaller frame for a cyclocross bike. They're a bit more versatile, but they won't be as fast on road as a road bike (unless you swap the tires), and they don't have suspensions like mountain bikes...

You can put thicker tires on road bikes, depending on the road bike... it depends on how much clearance the bike has around the fork and rear triangle to fit a thicker tire... some of them can take more, most can take a 700x23 instead of a 700x25....But many of them can't do much thicker than that.