The first one refers to the speed of the processing in relation to the length of the recording - so, say, you can process a 10 minute recording in 1 minute then you're 10x real-time. However, your analysis might require the full track to be available for best outcomes, and so you cannot really start with the processing until the full source is available.
The latter is what "online" processing refers-to, the ability to process on-the-fly in parallel to the recording. Obviously, this cannot be faster than real-time ;-) but hopefully it is not slower, either. Often times, though, you get a (somewhat constant and) hopefully slow offset, i.e., you can process a 10 minute recording online in the same time but you need another 10 seconds on top of that.
This is, by the way, not restricted to source separation, it applies to other disciplines as well, say, automatic speech recognition.