Last.fm mostly answers the question: "What I've been listening to mostly, this month?" whereas Digs.fm aims to answer the question: "Which album I discovered (i.e. listened for the first time) AND liked last month?".
This essentially comes down to the fact that Last.fm tracks "listens" of an album/track/artist, no matter if its the first time you're listening to it or the 1000th, where in Digs.fm _you_ (manually) track the _first listen of a release_.
Likewise, Digs aims to answer: "which album did my friend discovered and liked?" or "what album does my friend wants to listen to?". The premise is that, the fact that a certain person (which I know) found this interesting, is a good signal that I might be interested to that as well.
So, I'd say that, while they do share similarities, they're different tools that aim to tackle different needs. I myself was using Last.fm around 2008, then forgot about it until last year. I activated it, but I haven't looked at it ever since. I'd say it's a good tool to look at the yearly reports once or twice a year, but to me that was about it.
In contrast, I use Digs daily (I'm biased :P) to see what my friends discovered, add new albums to listen to my lists or pick something to listen from my "Want to Listen". Also, I love discussing about something that I listened or someone else listened (this is done by commenting on activity items).
Ultimately, I wanted a way to track and organize music discoveries (not listens), and make it so that I can easily share them with friends, and perhaps spark a discussion around those.
In fact, a few people asked for an integration with Last.fm, in which you could import the albums/artists from Last.fm in Digs, and add then manually add those that you want to a list.