I've done a bit of reading on the subject of USB mics and can offer a little advice..
First, it all depends on what you want to do. If you only want to record your playing, then any of the cheaper USB mics will do (eg Samson Meteor, CO1U).
If you want to overdub yourself over a pre-recorded backing track, then things get a little more tricky due to the issue of latency, which as far as I can figure out, is the time it takes for the computer to convert the sound picked up by the mic into a digital form and record it onto your hard drive. This means that your recording will be slightly out of sync with the backing track, so your playing will sound slightly behind the beat.
The solution to this problem is 'zero latency monitoring' - somehow the electronics in the audio interface can be made to correct for these differences. Some mic manufacturers have cleverly incorporated zero latency monitoring into their USB mics and provided a headphone socket built into the mic so you can listen and record in sync.
USB mics that have this system start at around £90 and include the Samson G Track, ART M One USB, Rode Podcaster and BLUE Yeti.
As far as I know, the Zoom H2 recorder doesn't feature zero latency monitoring, although it can be used as a USB mic.
Whether it's really necessary to go to the extra expense of buying a USB mic with this added feature is debateable - the latency may not be very noticeable, a delay of less than a tenth of a second may not matter as much with a saxophone as it would with a rhythm instrument and if you're really clever you could move the track forward in your recording software by a tiny amount to compensate for the delay, although doing this accurately would be very fiddly.
One of the main criticisms of USB mics is the low level digital noise that almost all of them have, it's something to do with picking up interference from the USB interface. While the sound from the sax is probably loud enough to drown out this noise, it may become more noticeable if you use compression or limiting to even out the levels of your recording.
For those people who don't have the money to buy mics and are stuck with using the built in mic on your laptop, the only option to improve your recordings is to use the EQ in Audacity (or whatever software you're using) - go to the effects tab and select 'Equalisation' from the drop down menu and you should find a graphic equaliser - turn things down a bit in the 1khz to 4 khz range and turn things up a bit in the 125 hz to 500 hz region and you should get something a little less tinny. Clicking the preview button will give you a 2 second sample of what it'll sound like. If you click 'ok' and end up with something you don't like, there's always the 'undo' function.
Although a lot of people on here like Audacity and it's ok for it's simplicity and lack of cost, it's difficult to use effects and processing easily due to not being able to listen to the results properly before you process the whole track. The stripped down version of Cubase (Cubase LE) that comes free with various recording interfaces allows a lot more control over your sound, although it's also more complicated to use.
I've not tried any of the freeware multitrack recording software that's out there like Studio One Free -
http://studioone.presonus.com/free/
or the ones listed here
Multitrack recording software Audacity free download - Page 1 , so can't say if they're any good.
Hope this sheds a little light on the matter and I've not made any glaring errors whilst typing after midnight