Beta has matchmaking but not a “Ranked Ladder”, meaning the matchmaking system pairs opponents at random. Without any system for checking skill level. If there was no matchmaking at all, then it would be like only being able to join games through the custom games lobby list
Classic has a Ranked Ladder currently, meaning new players match against new players. But if Reforged launches with no matchmaking, a lot of people are going to be unhappy… so I really really hope blizzard will keep the actual official “Classic” version of Bnet running for a while longer (unlike they did with RoC)
When they first announced Reforged, I vaguely remember hearing murmers about Classic profiles being transferred, and some sort of badge to identify longtime players, but Im not exactly sure if any of that will actually be true. I mean our profiles were linked, but I have a feeling stats will be erased and icons will be lost (just like they did to RoC players)
If they do shut off the original TFT servers, and there is no matchmaking available, it will drive a lot of players to private servers.
Its also important to note that Blizzard has struggled to make AT work with the Classic game, they claimed AT would be working last year but it never happened. Now in the beta they had it set so if you played AT you would match against RT players …(AT vs RT)… but thats a horrible way to play as its incredibly unfair for RT players, I really really hope they dont keep that in the final game.
They also told us over a year ago, that they were working to merge the servers, so that USwest/USeast and Euro players would be able to match against each other, but that isnt happening either. They did get it working for Custom Games, but Im 99% sure its not happening for Ladder which really sucks if it doesnt 
Not much else has been said about ladder though… some rumors came out last year about them changing the inventory system, but I think most people didnt like the idea, so Im not sure if they scrapped it or not (probably though)
Its definitely going to be a rough launch (similar to SCremastered) and it will probably take a few months before we get the full game like its supposed to be… hopefully