GRM is purchase price divided by gross rent.
Ex - I just purchased a property for 102,900 and the gross annual rent is 21,600. My GRM is 4.76.
102,900/21,600 = 4.76
COC is how long it'll take you to recoup your money in.
Ex - On said property, my all-in cash investment was 32,728 (down payment and closing). My cash flow after all expenses is projected to be $2,592. Therefore, my cash on cash return is 7.92%.
2,592/32,728 = 7.92%
Now here's the rub - both can and should be used when evaluating properties, but you need to decide what is most important. GRM to me isn't as important as COCR, but it's all relative to your goals.
Some people buy for appreciation, some people buy for cash flow, some buy only properties that achieve it all, and some unfortunately buy stupidly where nothing makes sense. Decide what is most important to you, use the metrics accordingly, compare those to other available properties, and then make a decision.
On this particular property, I wouldn't call it a home run by any stretch. But I needed to jump in, and I want to make improvements to the property to increase my monthly rent. I'm going for COCR and everything else is just a benefit.