A raw logic analyzer produces timing or state info.
With software they can provide a higher level view. In the case of RS232 instead of seeing the start, data, parity, and stop bits you would see the actual data transmitted as numbers and/or ascii.
When probing a micro you can use software to see traces processed by an inverse assembler. Given the right language package you can even see high level language statements included in the trace.
I have no experience with these affordable analyzers. They may include the tools so that you can even write code to post process you own traces. I feel this roll-your-own feature is more important then the ability to post process any given protocol.
EDIT: Having written the above it came to me that this may be way too much to ask of sub $1000 units.
Compare the speed in state and timing mode. Take a careful look at the triggering. It can make the diff between capturing what you need to see and not.