Bubbling Under is from another era, before streaming.
It was how radio schedulers could see what songs to playlist - if a song was charting on the Bubbling Under, it meant it had no radio play but was selling singles, so could be a hit.
Now we have Spotify and iTunes to tell us what's popular without radio support, and the Bubbling Under chart is less relevant.
So Trending 140 is more important. It tells the industry what people are interested in, in real time.
Bubbling Under, it means the song sold some copies and got streams and radio adds but weren't enough for the Hot 100. The Trending thing could be huge too since it shows you the twitter reaction to a song but it could also mean their fanbase is only tweeting a random song, so.