I've looked at both kinds of bikes and I don't see much of a difference in design and geometry. I want to buy an used Schwinn Passage bike and put some larger wheels on it and take it on some crushed limestone trails(flat trials). Would that be good?
Most cyclocross bikes have subtle differences in geometry that make the handling a bit more stable and often have slightly higher bottom brackets which is mostly a holdover from the clips and straps era of cyclocross.
But yeah, the big difference is clearance in the forks and especially around the brakes so that you can run much larger tires and that usually means cantilever brakes (or disc brakes or sometimes cyclocross specific V brakes) to clear those big tires and to allow enough room for mud clearance. The Schwinn Passage has standard road caliper brakes and most knobby cross tires won't fit unless you can find some super small cross tires like some 28 mm tires that might just barely fit with those brakes.
I ride my cyclocross bike on the road pretty frequently, often with the cross knobbies mounted but often with a pair of road wheels mounted and either way it's a fast bike on the road. BTW, the wheels really aren't larger, cyclocross bikes typically run 700c wheels just like road bikes but the cyclocross tires tend to be a lot wider like 32 or 35 mm tires instead of the typical 23 or 25 mm tires you'd find on most road bikes.
The gearing on many stock cyclocross bikes is lower than road gearing with compact cranksets being pretty common. For dedicated cross bikes the compact cranksets are usually set up with something like a 46 0r 48 tooth big ring as opposed to the 50 tooth rings you'd find on a compact equipped road bike and the rear cluster is usually something like 12-27 instead of more aggressive 11-23 or 11-25 cassettes but all that stuff is interchangeable. Similarly a lot of cyclocross bikes are built with in line bar top brake levers which are nice when riding off road but again you could add those to a road bike to have multiple braking options.
My cyclocross bike is by far my most versatile bike that's useful on road, off road, for long days in lousy weather and even for the occasional fast road event when my race bike is out of commission. Basically if I could only have one bike it'd definitely be a cyclocross bike.