The 24-pin ATX is a later version (2.0 I believe). Earlier versions can work with it, but only with yet another adapter. Once again, check inside YOUR computer to see if it has that extra power connection near the standard ATX connection. If not, you probably won't need the Dell to ATX converter. HOWEVER, since you got an earlier version ATX PSU, you need a converter to get it from 20 pins to 24 pins:
So, here's a recap:
1) Find out if you need the ATX-Dell converter by looking for that extra connection coming from the PSU. If yes, go to step 2. If no, go to step 3.
2) Purchase the ATX-Dell converter and attach it to the motherboard. This will leave just a 24-pin connection to plug into.
3) Purchase the ATX 20-pin to ATX 24-pin converter linked above. If you needed to follow step 2, plug one end into the Dell to ATX converter and the other end into your PSU. If you DID NOT need to follow step 2, just plug one end into the motherboard ATX connection and the other end to your PSU.
5) Damn Dell for being so complicated.
6) Rejoice again.
7) Make pie.
8) Eat pie while rejoicing.