IMHO, the largest change between the playtest multiclassing rules and the Basic multiclassing rules is the need to meet the ability score requirements for both the former class and the new class. This is huge. It might also put a strain on people's abilities, in both senses: with a 27-point-buy rather than a 30-point-buy, it will be harder for PCs to have scores of 15+ in two different abilities, and that will limit their players' abilities to multiclass.
For that reason, WotC might have cut down the requirements: instead of 15+ (or 13+ to two stats), the PHB requirement might be 14+ instead. We'll see.
The Basic PDF file clearly states, "When you gain a level in a class other than your first, you gain only some of that class's starting proficiencies."
What makes the most sense? A new Wizard would need to know Arcana; a new Fighter would need at least Medium Armor and Shields, maybe a couple of weapons (but not the entire weapon list).
I'm thinking the multiclasser might be viable with 1 new Skill, 0 new Saves, all new Tools, 2 new Armors, and 2 new Weapons. I doubt that will turn out to be the actual rule, but it makes sense to me.