If you're going to record streaming audio, that's the wrong way. You would be going D/A, through an analogue amp, through an analogue cable, and then back A/D in the recorder. Total Audio Recorder and similar products will record the stream digitally to your hard drive in wave or any of several other format, yielding much better quality (as well as more convenience).
Internet feeds are compressed. Some are highly compressed, like to 20kbps and sound absolutely awful. Some that are at 128kbps or higher sound pretty decent, while others do not; there are more factors at play than simply the data rate. The quality varies by a very wide margin among streams.
A good FM transmission on a good tuner sounds much better than any internet stream I've ever heard. However, a bad FM transmission on a good tuner, or any transmission on a sufficiently crappy tuner, may not sound better than a good quality stream. Although I have a nice tuner, there are a couple of syndicated shows that I listen to or record from an internet stream because I don't get very good reception of the local station that broadcasts them.