Exlicitly, what model of the athlon II or i3?
As a rule, the i3 is considerably faster.
Look at this link, enter your cpu's and look at the tasks you will do the most.http://www.anandtech.com/bench/Product/143?vs=121
Exactly what are the specs of the proposed systems? Of your old systems?
I suspect that if you are on a budget, you can assemble a better system yourself for less. Particularly if you can reuse some of your old parts.