Actually, performance has nothing to do with how many cores you use. What matters is the architecture. I ll try to explain as simple as I can: Even if only one core of your athlon was used it would still be faster compared to your Pentium 4. Thing of it this way: The athlon is designed to execute 2 instructions per clock cycle (2x2.3 Ghz) while the Pentium can only excute 1 (1x2.8Ghz). Do the math.

It isn't working exactly like this but you get the big picture.
More cores mean better performance when multitasking or when using programs that can utilize multicore CPUs. Not all programs are designed to do this.
My final suggestion, now that you know how these 2 CPUs compare: If you plan to use the computer as the family machine keep the second. If you want it just for the kid then keep the one that you like more.

If you are certain about the hardware demands of the programs that will be used and all of these programs are light then the performance difference may not be noticed.
Since a child will use the computer keep in mind that the second machines will run some older games faster. The first machine may not run some games at all. Of course neither of these machines will run newer games.