This is a great question!
mbtile is a filetype that can contain either raster or vector tiles. Think about it like an Adobe Illustrator .ai file -- it can either contain vector or raster data. Both raster tiles and vector tiles can be in mbtile format.
With TileMill, your vector data (shapefile, CSV, etc.) was combined with your CartoCSS style and passed to Mapnik to generate a bunch of tiled images. You could then store your tiles and feed them to a web map via a tile server. With Mapbox Studio Classic, you are converting your data to vector tiles and creating a CartoCSS style to style the data, but the image tiles aren't created until the map is requested via the browser. Ultimately, the output on the map is the same -- a grid of raster tiles.
The interactivity you created in TileMill with raster tiles is happening via a UTFGrid. This is an invisible raster layer that sits beneath the image tiles and adds interactivity to certain locations. The interactivity is not actually connected to any individual features at all -- it's just at the same location as your features. This is also possible in Mapbox Studio Classic, as the output in the map will also be a grid of raster tiles. You can read more about how to do this in Mapbox Studio Classic in the style quickstart guide.
So to answer your question, the move to Mapbox Studio Classic was to take advantage of vector tiles. They are a lot smaller than raster image tiles, and they can be styled on the fly, so, for example, you don't have to store two separate sets of images to accommodate retina screens. It also means you can have multiple styles applied to the same vector data on the fly, which means you don't have to regenerate a whole set of images for every style or every small change.
