Well, the Ithaca 37 is built on the same design as the one Remington used two steps before the 870. Originally it was going to be the Model 35 but they had to wait two years for the patent rights to expire. It is a solid well proven design, but requires more machining and handwork to complete than the 870. Some people prefer the feel and/or bottom ejection of the 37, and it is a lighter gun. The 870 is simpler and more robust in my opinion, and is the most popular shotgun model in the history of the world. I do not like bottom loading through the magazine on the 37, but that is a purely personal preference issue.
If I was going to do a lot of carrying and didn't prefer the Remington's ergonomics, I'd look at the 37. If I was looking to shoot a lot of heavy loads I'd go Wingmaster. I am not into the whole tactical thing, content to rely on one of my duck guns for home defense if the dogs and the 357 aren't enough, but there is probably 50 times more "stuff" available for 870s than 37s. The older 37s required factory fitting to swap barrels, whereas even Mossberg makes some extra barrels to fit 870s.
Either gun will work.