I have owned and commuted on both and while they may seem similar they are actually distinctly different bikes. In my experience, the LHT is better bicycle both in general and for commuting for the following reasons:
The LHT is designed to carry weight and has thicker, stiffer tubes and lower bottom bracket that provide stability with only a slight sacrifice in weight. On the other hand, the CC felt unstable when loaded, especially in the front end.
The LHT has more relaxed geometry that is better for commuting but is still quick and responsive enough to make it fun to ride. The CC has awkward geometry that felt uncomfortable with a longer effective top-tube than the same size LHT.
The LHT comes with lots of useful brazes and has nice touches such as fancy fork and rear dropouts and extra spoke holders, while the CC doesn't even have a cable stop for the rear brake! It also comes with nicer components than a CC, with XT hubs and rear derailleur.
While I'll allow the CC is a nice bike, the only thing that it has on the LHT is the horizontal dropouts which allow single or fixed gear riding. However, this is only an advantage if you always ride SS/FG, as changing back and forth is too time consuming to be done on a regular basis.
I have since sold both bikes and am now on my third Surly machine, a Big Dummy. If your interested, I could give you another 20 reasons why a BD is better than a CC or an LHT!