I have no idea what you mean by board game; those are as varied in mechanics and execution as tabletop RPGs and video games. What's a board game like experience? If you don't want to answer me, that's no big deal to me and I can just leave the discussion. Role playing, social and exploration stuff is still done in the exact same way as previous editions (although with rituals available to everyone, magic users weren't the sole holders of the trump card there). It's primarily combat and pacing rules that changed drastically, and that was to fix issues in previous editions, primarily 3E, such as boring (or worthless) healing in combat, the 5-minute adventuring day with spells, boring or a lack of downtime rules (by cutting out mandatory downtime entirely), easier monster and encounter creation, etc.
I can't relate to your observations about the changes between AD&D and 4th Edition "mostly being the players' options all being in the PHB, and not some in the DMG or even in the MM", first let me say. I didn't know how to answer you exactly, but you give me more of an opportunity here with asking what I think is in a board game like experience.
1) characters, or pieces representing your place, move from square to square on a board
2) all players take equally significant turns
3) there are clear objectives
Some of my observations have been the "Dungeon Tiles", for example, which are like a changing board. You reconfigure the pieces for each room where an encounter takes place. It's very different than imagining your way through the dungeon, and keeping a map, but of course there could be outdoor encounters.
The vernacular is number of squares, and there are new board game like rules about sliding, pushing, and pulling. These are perhaps not meant to be taken this way, not by someone who isn't thinking of D&D as a board game, but they are fitting for a board game and deliver a similar feel.
The players taking equally significant turns point is best shown with the combat roles themselves. Teamwork is focused on in new ways, and the balance you see in 4th Edition establishes a hitherto unseen parity between all characters and participants. This permits a board game experience. In a traditional D&D game, characters are not balanced in this way. They are still balanced, of course, but I mean in different ways. The goals are to balance them in such a way that they aren't too powerful as individuals in their world, not so that everyone playing in the same game has an equally important character at all times.