There is a huge difference between both.
In general terms a Loadmaster is responsible to get as much cargo as possible on the airplane while the flight dispatcher is responsible to make a flight plan to get the aircraft from A to B.
A loadmaster is part of the flight crew and is usually listed on the gendec. There isn't a federal license or anything for loadmasters, those are licensed by their companies. Loadmaster jobs can be as easy as just trimming a Fedex airplane with containers containing to complex endeavours like special loads and stuff like loading whales on an aircraft.
I can't really speak for other companies but where I work the motto is 'you name it, we fly it'. We are specialised in heavy, oversized cargo which makes it very interesting. When we plan flights with such cargo it becomes a big game of Tetris where you work yourself up to the max capacity of the airplane. In my case 747-400F & -8F. Things to consider are obviously getting the freight to fit through the nose or side cargo door, floor limits, asymmetrical loads etc. Another important factor are attachment points for straps.Example: A 30 ton blow out preventer can't be properly secured in the tail section of the aircraft.
Once we are done with our trim we send the ZFW to flight dispatch who then uses our numbers to crunch the fuel requirements and their flight plan. Sometimes the airplane is too heavy for specific routes so they have to adjust, or the aircraft will be too heavy to land etc. That's the job of dispatch to figure that out. Now we do work hand in hand so sometimes we coordinate early with them so that they can provide us with a max ZFW so that we can plan accordingly.
A flight dispatcher could work the same yes. During the training you will perform some weight and balance calculations. Generally for passenger operations flight dispatch and crew handle that part. Cargo is very different and if you go to special loads it becomes rather complex which is why is is split up. Besides the fact that a loadmaster usually is part of the flight crew so we go with the airplane to ensure proper handling of the freight at stations etc.