Keep in mind that RV lifestyle is one we have chosen for about the last 35 years....so of course I'm going to be bias. ;)
That said, the weather in May in Yellowstone would be my biggest worry for traveling in an RV. It could easily snow and then you might be stuck a day or three. I'd also worry about the cold nights and freezing temps that can burst water pipes in the RV. Depending on the snow fall for the year, roads like the Beartooth might not be open yet.
The Good:
I remember driving the Beartooth Highway and stopping at one of the little overlooks, cooking breakfast with the most breathtaking views! Cooking all our meals and saving tons of money! Cheap camping if you stay in the park (the exception will be Fishing Bridge with full hook ups) If one of you is 62 or older, you can get the Senior National Park Pass which is a one time $10.00 fee. This gets you into the national parks and up to 50% off on camping. Yellowstone is one that gives you 50% off. Not having to pack and unpack a suit case every time you need to move to a new location. A bathroom where ever you go!
The Bad:
Private campgrounds can be expensive especially near Yellowstone, large cities or other major attractions. We get about 7 1/2 MPG in our 30 ft. RV...didn't get much better with the 24 ft, but it was a 1985 model. You have to dump the waste tanks. You have to break camp every time you want to drive off. (This wasn't so much an issue in our smaller motorhome. We didn't keep much out and if you don't have hook ups, it's pretty fast) If the RV has slide outs, then you have to put those in.
You didn't mention what size RV you were going to rent. We took our 24 ft motorhome there several times and never had an issue with finding parking. The year we took our 30 ft motorhome we came home and bought a tow vehicle because we couldn't find places to park with the larger rig. Some of the minor roads & smaller parking lots in the park have RV size limits now.
