The disadvantages of 5870's are that you can fit less hash/s on a single motherboard and that they are less energy efficient compared to the 5970s.
I doubt about energy efficiency, it's only few percent of difference because in Windows at stock frequency 5970 has only 1.5-1.7 times more hashing power than 5870 and consumes 1.5 times more watts. When you overclock and overvolt 5970 it will take considerably more watts than it's specified <300 W.
5870's default frequency is 850 Mhz and it usually runs fine at 950 without any modifications or special cooling.
5970 is clocked at 725 Mhz and usually needs voltage rising to run faster than 800 Mhz. You'll also need serious PSU and cooling.