If you wanted to test your system's videocard and stress its vertex and pixel shader perfomance as well as direct x 9 perfomance use 3dmark2003.
If you want to test the general speed of your whole system together including Memory, CPU, Motherboard, and Videocard use 3dmark2001. 2001 is more accurate in terms of benchmarking the system as a whole as compared to the 2003 model. Also in 2003, both companies i wont mention have been accused of cheating so there is some concern as to its credibility in the public's point of view.
Your question however might be a bit unfair due to the fact that 3dmark01 came out 2 years earlier and has gained popularity during this period. However, 3dmark03 is fairly new and hasnt had time to generate as many users partially due to time contraints in comparison to the old version.
But I would still say if they were released at the same time, 3dmark01 would be a more popular indicator as it tests the system as a whole. The disadvantage of using it today is that it does not test any direct x 9 game engine, and only 1 direct 8x engine with the rest direct x 7. So its not very indicative of the true current world performance. But most assume if direct x 7 and 8 results are faster with 1 videocard than the other in '01, then chances are 9 will be faster as well (not always true though).
For best understanding of the gaming performance of your computer use game benchmarks.
Cheers