HERMS - uses a heat exchanger (usually residing in the HLT), mash is circulated through this heat exchanger and deposited back into the MLT. In my case, it's gently returned on the top of the grain bed, doesn't really disturb it much.
RIMS - the wort is circulated to another vessel or RIMS tube and heated while in that vessel. It then returns back to the MLT.
The main difference is one uses a heat exchanger to change the temp of the recirculating wort while the other is heated without a heat exchanger. A RIMS is usually able to perform much quicker steps at slightly higher risk of scorching and the need for another vessel.