Difference between road & mountain drivetrain Basically MTB and road bikes have different brand parts  (like deore / XT; Sora / Utegra etc).  

What are the main differences between the 2 types, that will make it incompatible?
Are there things that do differ, but not affecting the whole system, so I can change only one of them to improve my commute?
I think this is quite a broad question, so I'll highlight a few parts of the bike:

Seatposts/saddles: Probably interchangable

Forks: You wouldn't want to swap them, and chances are a beefy mountain fork wouldn't go into a road frame anyway, even if you could get the right headset.

Gearing: Road bikes typically have higher gearing than mountain bikes 

There are larger chainrings on the crankset on a road bike - typically 52/39 or something or a 50/34 versus a big ring in the low 40s or 30s on a mountain bike. 

The rear cassette is often smaller on a road bike as well (you won't see a 36t big rear cog on many road bikes, unless they're touring bikes).

Front derailleurs: FD's need to be shaped to the number of chainrings (typically 2, less so common 3 on a road bike vs often 3 on a mountain bike) as well as the size of the chainrings. For this reason (and in some cases the amount of cable pull needed to move the front derailleur) you can't mix road cranksets with mountain front derailleurs and mountain cranksets with road front derailleurs. On a mountain frame, you probably can't clear a road double if you were to mount it on, and you may have derailleur mounting issues when putting a mountain crankset on a road frame in some cases. 

Rear derailleurs: Until 10 speed, you could swap them around when you stuck within the brand (then they introduced different cable pulls introducing shifter incompatibilites, so road shifters have to be paired with road derailleurs and similarly for mountain). The mountain rear derailleurs can shift larger rear cogs than road ones.

Cassettes: You can interchange them if you want in a lot of cases (touring road bikes often use big rear cogs). 

Shifters: The front derailleur needs to be matched to road/mountain shifter. Rear derailleur will need to be matched if >=10 speed. Friction shifters always work.

Wheels: Mountain rims are often wider to support wider tires at lower pressures and are usually built to be more robust. Your hub widths are often the same, but mountain axles can cause compatibility issues (not just your usual QR) with different frames. Note 700c = 29". Often wheels built for tougher road riding (e.g. touring) use mountain bike hubs. 

Tires: Mountain tires are typically fatter than road tires. 

Cables/Cable Housings: You can use the same types.

Handlebars: Mountain and Road bars have different diameters, so brake levers and shifters designed for one will not fit on the other.

Brakes: Typically, road brakes (mechanical disc brakes for road, caliper brakes, cantilever brakes) are short pull while mountain brakes are often long pull (v-brakes, mechanical disc brakes). This means you require a travel agent or special levers to run v-brakes or certain mechanical disc brakes on a road bike. Hydraulic systems are quite new on road bikes. Since you can't mount the levers onto a mountain bar, exchanging them is somewhat moot at this point. 

Racks+fenders: These are wheel size and disc/not disc specific. Read the directions on the rack+fenders. 

Generally, road stuff is lighter than mountain stuff. Probably less durable in a crash though. 



Now to part 2 of the question: What can you change on a bike to improve your commute? This question has been asked on SE a bunch of times, but typically the answer for commuting on a mountain bike is to lock out your suspension if you can (or switch to a rigid fork) and fit slick tires. You could also try a smaller cassette if you want, but other than that, you're going to want to get a new bike. If you have a road bike, you sometimes will want to fit a big cassette so you have lower gears available (you may need a new chain, and to make sure all the capacities of the derailleurs are within spec, else you may need a new rear derailleur) and bigger tires to make things more comfortable.