Either frame will work fine for your purposes. Fortunately, that means that as long as the bike fits, I don't think you will go wrong.

The Cross Check will have a slight edge with gravel or dirt, especially since the stock config includes cross tires. You might want to lower the gearing a little bit. The crankset is very versatile -- you can set it up as a standard double, compact double or (with a different BB) a triple.

If you go with the Cross Check though, I'd be careful with the fit. Due to the geometry, you would normally go for a smaller size frame than with a road bike; e.g. if you ride a 52cm road bike, you'd want a 50cm Cross Check.