First of all, your question's title is misleading, because your question is limited to Mini DisplayPort and USB as sources. If your question were general, the answer would be different.
The concepts of the two solutions are as different as they can be.
Mini DisplayPort to HDMI adapters will mostly be cables which map pins. This means that this is a passive device which just maps one connector standard to another. Your computer's graphics card will be used.
USB to HDMI "adapters" are really USB graphics cards which require a driver. The computer will transfer the resulting image data as video buffer differences (or similar) to the graphics card. I've researched some benchmakrs for another use case and there is a significant performance drop. It's alsways okay for office work but you should check test reports of specific devices against your requirements. The video modes supported by the USB graphics card may be different from those in your computer. That may be good or bad.
Having said that, it's questionable why you would want to go for the USB solution in your case. The data transfer on USB as well as the power supply for the external graphics card via USB will at least lead to shorter battery life and will also mean some work for internal buses.
The USB solution may be the best choice if you're dissatisfied with your computer's graphics card and don't intend to play 3D games.