The first sentence is incomplete; we would use this construction to talk about specific health problems e.g. "...many of the health problems encountered later on in life". More information is needed because we have used the determiner "the" to indicate that we are talking about a specific kind of health problems. The second sentence refers to health problems in general. Hope that helps!

I would say in general little difference. I think of "many" as more general and to be slightly more specific with "many of". In a sentence like the first I would add like in the world today or in a certain country to specify a little more of the scale or location you are thinking on. I hope that helps.