As a gauge of overall system performance in business and content creation applications, SYSMark 2000 puts the 1.1GHz Thunderbird at about 5% slower than the 1.13GHz Pentium III. As you can tell by the scores above, SYSMark 2000 greatly favors CPUs with a fast L2 cache, as well as those CPUs that have a large L2 cache. Case in point would be the clear 10% performance advantage a 1GHz Thunderbird with its 256KB on-die L2 cache running at clock speed holds over an equivalently clocked K75 Athlon with a 1/3 speed off-die 512KB L2 cache.
You'll also notice that the Duron, clock for clock is about 3% slower than the older K75 Athlons and about 12 - 15% slower than the Thunderbirds again, on a clock for clock basis. Compared to Intel solutions in the same price range, primarily the Celeron, the Duron completely dominates because of its [effective] 200MHz FSB and 100/133MHz memory bus compared to the Celerons 66MHz FSB and 66MHz memory bus by default. Even when overclocked, the Celeron is still noticeably slower than the Duron on a clock for clock basis yet is comparable in terms of price. The only factor that is holding the Duron back here, and has apparently been hampering sales is that Socket-A motherboards aren't as prominent as Socket-370 solutions and they are noticeably more expensive than some of the cheaper Socket-370 boards.
From an Athlon vs Pentium III comparison, the performance between the two chips (Thunderbird vs Coppermine) is very close, but when it comes down to price, the Thunderbird is the more affordable solution.