They're not EXACTLY the same. Look at this page, and scroll down to the Basic / Expert rulebooks. You'll see that there were some small differences, especially in the Expert rulebook.
There are quite a few differences between the 1981 and 1983 sets, the majority of which are in terms of format, art, writing style, etc. The rules are mostly the same, certainly very compatible, but there are subtle differences. I think the 1991 Rules Cyclopedia had more changes in terms of things like the thief skill percentages at different levels or saving throws or XP or the like - taking into account the higher levels of the later Mentzer boxes. In my experience you can pick any of these three and play and use the various modules and accessories without issue. Maybe it's me, but I find less difference between 1981 and 1983 basic, than I do between 3.5 and Pathfinder. Naturally, the latter games are also more detailed and complex leaving more room for differences.