I think Dr. Martens are usually a bit chunkier, more robust looking, and they often are actually a little bit heavier on your feet. They have a fun selection of styles, but not really many dressy options, that's the biggest difference I see between them and Miz Mooz.
I know many people who have had their Docs for years and years, the quality is outstanding.
I love lots of Miz Mooz styles too and have a few myself. In the past I didn't always find them super comfortable, but that has improved a lot with the last few collections. They have dressy and more casual options and often shoes in fun, bright colours. I haven't had mine long enough to say how durable they are, but so far, so good!
Prices for both brands are quite reasonable, in my opinion.
I definitely wouldn't hesitate to buy again from both brands.