Well technically one is a public land hunting and fishing advocacy group and the other is a shooting sports advocacy group. The divergence makes sense, NSSF doesn't really care about public lands they care about 2A rights and are going to kowtow to the current administration, who they feel is pro 2A. Net net this is a bad move for public lands hence BHA's response.
The reversal would be true as well, BHA is going to back candidates and administrations that are pro-public land regardless of their 2A stance.
If there were no public lands BHA would be out of business but NSSF would we going strong (think Texas), if there were no guns BHA would still be going strong (archery and fishing).
Point being neither of this groups is anti-hunter, they just have different mission statements and are focused on different things. There is no reason why someone couldn't support both, you just have to be honest with yourself and read between the lines.
Trump does not care one iota about hunters, neither do any of the democrats running for president, therefore we need to lobby for our interests hence it's a good thing to have both of these groups around.