The custom has a taller and narrower front wheel, which I think looks better than the classic. Im use to dirt bikes that have big front wheels, so thats my mental image of what a motorcycle is suppose to look like.
The custom has pegs, the classic has floorboards. You cannot stand up on the pegs like a dirt bike, or a sports bike, your feet end up in about the same place on the classic, but the mounting holes are different, so you cannot just bolt the floorboards from a classic onto the custom. It might actually be a different frame, just because of those two mounting bolts.
One nice thing about the custom is the half cut off rear fender. To take the back wheel off you only need to raise it an inch off the ground, and the tire will pull out the back and clear the fender. With the classic and its full rear fender you have to either jack the bike over a foot in the air, to get the tire out from under the fender, or take off the passenger seat, drivers seat and half the fender bolts so you can swing the fender up in the air, with the tire only an inch off the ground.
Other than that, the bikes are identical.