I have both a LHT and a CrossCheck. I've toured and commuted on both. here's my takeaway:
Either bike works fine for general around-town riding or commuting. I notice two general differences between the bikes. First, I built the LHT to have a more upright position which I find more comfortable, especially on rides any longer than an hour. There's nothing that says I couldn't setup the CrossCheck to have a very similar position.
Second, the two bikes ride differently. The crosscheck's steering is more lively and the frame feels more rigid. so like, when i stand up to hammer out a hill or sprint to the next traffic light, it FEELS like every bit of energy is being transferred to the rear wheel. The Long Haul Trucker FEELS way different - I hardly ever stand because it feels useless on the LHT, like the bike is telling me to sit back down. It's a less lively ride but much more comfortable. In practical terms, neither gets me to work faster than the other on a consistent basis.