It's hard to describe the differences in quest progression and flow on the beta for Warlords. Although the test servers are currently riddled with players, which means they are also riddled with extreme amounts of lag here and there, it's still possible to get an overall idea of how the quest design and flow has changed from Mists ... and there are some major changes afoot.
First and foremost -- quest items are no longer carried in your bags. If you have to pick something up, it's tracked in your quest tracker, not taking up space in your inventory. This means far less clutter as you're running around completing quests, and much more space for loot. In terms of space-savers, this is an absolutely wonderful change, and drastically cuts down on the need to head back to town and find some sort of vendor mid-quest chain. Although at least in Shadowmoon Valley, that doesn't seem to be a very large concern either. Vendors are all over the place out in the world -- there are draenei camps dotted all over the zone with vendors who will happily buy whatever you're selling. Whether or not this carries over to other zones, I have yet to see.
Solo scenarios have made a return, but they are seamlessly integrated into the questing experience. You no longer have to queue for a solo scenario. If you are in the correct area and at the correct portion of a quest chain that includes a scenario, it will automatically phase you out from the rest of the world, and the scenario will begin. No loading screens, no menus to deal with, simply walk into the area and you'll see your scenario objectives appear just above your normal quest objectives.
Exploration is openly and actively encouraged. Scattered around the zones are bonus objectives that offer cool rewards -- loot, followers for your garrison, gold and XP. These bonus objectives aren't offered by an NPC, they simply begin when you walk into an area. Leave the area, and the objectives will fade out. Head back into the area, and the objectives will pop back up again -- and the UI will keep track of just how far you've gotten with that objective. You can duck out mid-objective and come back later if you'd like, or never re-visit the area again. In addition to bonus objectives, rares are back in full force, and far more easily tracked on the minimap. If you've ever spent time on the Timeless Isle, you'll instantly recognize the minimap icon for a rare spawn mob.
These may all seem like small improvements, but when you add them all together, you have a leveling experience that feels much different than anything we've had in WoW before. Sure, they're simple quality of life changes, but each tweak to the system is another moment in which you don't have to turn your attention away from what you're doing -- you can concentrate on the story being told. And if you don't care for story, you can concentrate on just getting the job done, without a lot of side trips or shuffling through UI menus. Overall, it's a lot of little subtle changes that add up to a much more pleasant experience -- we'll have to see how the other zones compare as beta continues.