Without referring you to another website/chat room, the differences are the mileage and the difficulty.
The main trail is 11 miles each way, the MR is ~6.
Main trail is class 1 all the way. MR involves some Class 3.
Main trail will probably still have snow on the switchbacks July 1st so you may want to bring, at a minimum, microspikes. They sell some nice ones for a good price at the Portal and also at the Hostel. They also sell maps.
MR also involves utilising the E-ledges that time of year and if you're not comfortable with that I would suggest sticking to the Main Trail.
I believe there is a smaller daily permit quota on the MR than the MT.
I hope you find this helpful. Bring mosquito protection.