First of all, the road bike won't make you sweat less. What will make you sweat less is slowing down, not speeding up.
Second, before considering the possible speed advantages of a road bike, set up you MTB for speed. Putting on slick road tyres will make a big difference. Lock out any suspension.
Once you've done that, the advantages offered by road bikes are aerodynamics and weight. The former is the more important, riding in the drops saves a significant amount of power for a given speed. Weight matter most on hills, very little on the flat, and maximising the benefits costs a lot of money. Generally speaking, light=expensive. And at the end of it all, the difference in the speed of your commute won't be huge. On my most recent commuting route, which was about 16 miles, if I took the touring bike with panniers containing clothes, laptop etc. - about 45lbs in all - it took me 65 minutes. If I took the full carbon road bike, carrying nothing - about 17lbs in all - it took me 55 minutes.