I took over maintaining Quest a year or so ago, so I cannot claim to be objective here, however…
Quest is much more popular. That does not mean it is better, but it does mean the forum is much more lively, and there are a lot of people willing to help out. Check out the forums and see what they are like.
Quest games, whether created on-line or on the desktop version, can be uploaded to the textadventures.co.uk website, which I would say is a major repository for text adventures (not just Quest). As far as I know, users of ADRIFT games need to download the runner, which severely reduces the amount of plays. I released a Quest game, Deeper, Jan 2016, and it has had 26384 plays and 2194 downloads. Compare to Top Hat, the top rated Adrift game, released in 2009, with 1670 downloads. I would suggest Quest games get played much more.
The last time Adrift was updated was June 2016. Quest was last updated July 2018, and will get further updates.
The mapping in the Adrift editor is cool. You can create a map in Trizbort, and export directly to Quest, but it is a one-way deal.
There is a new version of Quest hopefully on the way as someone mentioned, but the old version will still be supported. The webplayer supports Quest 4 (Quest 5 dates from 2011), and I think supports Quest 1 still, though I am not aware of any games that old (Quest 1 to 4 were developments of the same system).