They are fundamentally different in that one is freeze-dried and one is dehydrated. I am assuming we are talking #10 cans for both, since this is long-term.

Mountain House makes some basic ingredients (meat, cheese, vegetables) but they specialize in complete freeze-dried entrees. If you want very simple food prep, this is the way to go. Scoop food, add boiling water, wait 10 minutes, eat. Of course, it's more expensive, but it makes a great quick meal. The only problem is that everything is pre-spiced, sometimes heavily so, and you either love the taste or want to use it as spackle. The pouches are a good way to try out various dishes to see if you can eat them regularly.

Provident Pantry (which I have not tried, but I have used Rainy Day Foods, which looks very similar) is dehydrated, so it is more individual ingredients: rice, beans, TVP, tomato powder, pasta, fruit and veg. Dehydrated stuff requires some simmering and cooking (hence fuel) from scratch, and you'll find more dried mixes like for pancakes, breads, etc. The upside is that you get to season it yourself and vary the menu more than with freeze-drieds, and it's much cheaper. If you know how to cook even a little, you can make a lot of "normal" meals from dehydrated stuff.

I have not yet built up my stock to where I want it to be, but right now I use a base of dehydrated food with a few select freeze-drieds for quick meals and areas that dehydrateds don't cover (meat, cheese, some fruits like berries). It works out well, and I mix it slowly into our regular menu so if we ever need to use it exclusively there will be no great shock.