Grob My last aeroplane was an L4 65hp Cub and my current mount is an Auster Autocrat with a 100hp Cirrus II so I'll try to expand the differences already mentioned by QDMx3 and Kingy.
THe original Auster was indeed a licence built Taylorcraft. However the ARB (forerunner of the CAA) reckoned it was too flimsy. The makers basically kept the same design but went up a size in all the tubing wall thicknesses etc. This gives the Auster a much more solid feel than the Cub. They are classically British in their over-engineering. Indeed they feel like they really were engineered to meet a requirement whereas the Cub feels like (and indeed was) built down to a price. Over the years there were redesigns around the basic airframe but with the exception of the AOP9 and the one-off AOP 11 they were pretty much the same airframes but with a variety of engines.
The Auster can be more of a handfull on the ground in a cross wind as most have free castoring tailwheels (what prat thought that one up) whereas Cubs have tailwheel steering. However, a lot of Austers (including mine) have had tailwheel steering retrofitted.
Landing the Auster is slightly more challenging than the Cub because, IMHO, the undercarriage is stiffer and the tyres are harder. This gives a harsher feel to the landing but it's not that much harder than a Cub. Most of the groundloops etc were caused by the aforementioned tailwheel.
In the cruise the Auster bops along at 95+ mph against the Cub's 70-75. That 20 mph makes a big difference if there's any kind of headwind. I disagree that the Cirrus is rougher than the Conti A-65, well, mine's as smooth as the A-65 in the Cub was and I've checked the balance on the props on both Cub and Auster and they were both OK. The engine is more maintenance intensive than the A-65 as you have to adjust the valve clearances every 25 hours. The Auster is definitely noisier as it has no silencer/muffler. Mine will soon though cos I was going deef!
To sum up after all that, most people think the Cub is more cute than the Auster and that's a factor in the price equation. The other factor is that spares for the Cub are easily available, including the engine, at a reasonable price. Auster airframe parts are available, also at a fairly reasonable price but engine spares, especially for the Cirrus, can be problematical. They can however be re-engined with the Lycoming 0-320 or 360 if required. Another factor is that they were the ubiquitous trainers of the 50's & 60's and were common as muck then. Finally, they suffered from the reputations of the later overweight Beagle models.
Personally I prefer the Auster as it is more robust and, although not as "chuckable" as the Cub due to higher control forces and not very effective ailerons, it's still loads of fun to whizz about in. My aircraft, if fitted with the correct seats, can be looped, rolled etc legally. It is stressed to +4.5/-2.5 g compared to the Cub's +3/-1.5 (I think). A further bonus is that all Austers have a perspex roof (some are fitted with dinky little blinds). The L4 has a clear panel roof but I find the J3's claustrophobic (I am 6' and 15 stone!) with their solid roof and small rear side windows. Although you can open the side of the Cub up in flight, you can slide the side windows of the Auster back as in the Mark 1 Minis and stick your elbow out for posing purposes
And finally (I can hear the sighs of relief), Auster is what the Romans called a light southerly wind. It was felt during WW2 that this fitted well with the likes of Hurricane, Typhoon, Tempest etc.