They are similar but not the same .Entitled means " to have the right to do/have something". That doesn't mean that in reality men have the days of maternity leave they expect to have.

The first sentence asks how many days men are entitled to take, or in other words, that they have a legal right to take. The second sentence could have the same meaning depending on the context of the conversation, for example if you had just been talking about legal entitlement for women. Or the second sentence could mean how many days do men typically take in their country (which could be different from what they are entitled to). Both are ok. The second sentence is probably slightly more casual but still absolutely fine.