This is an interesting one. Specs wise, they both trade punches with GT440 hitting with the higher core clock and the 6570 making up for it in every other aspect.
In the end, the chart by our very own Tom's Hardware seems to name the HD6570 the winner.
Both are relatively low power cards so, I don't think power will be that much of a problem. Unless of course your PSU is seriously dodgy.
Don't expect maxed out graphics for your games but they'll serve you just fine.