If you're looking at a relatively low speed protocol where the physical layer is (assumed to be) clean and reliable, then a cheap USB logic analyzer will often be good enough. I have a whole collection of these and for the most part, there isn't much difference between them.
In most other cases, a scope has significant advantages for serial decodes, most of which other people have already touched on, such as hardware triggering. One of the biggest ones is the mixed signal (analog + digital) aspect.
For example, if I'm trying to debug an issue on a CAN bus, I might want to only capture frames with a bad CRC or a recessive ACK bit. This requires a hardware trigger unless I want to somehow try to capture and post-process a huge number of frames (and assuming the error is frequent enough to make this practical).
But having captured an errored frame, how do I determine the root cause of the error? Was the CRC incorrectly calculated by the sender? Or were bits corrupted on the bus? If I trigger on errored frames, I can then look at the analog signal(s) to see what happened during the errored frame - noise, misbehaving node, etc. This is what makes a mixed signal oscilloscope such a powerful tool: the ability to correlate events in two domains.
There's a reason why pretty much every modern oscilloscope - even "hobbyist" scopes - has an MSO option, even though this option is always considerably more expensive than a $20 USB logic analyzer.
Again, there are plenty of cases where that $20 USB logic analyzer is more than sufficient, but there are also plenty of cases where it isn't
