Tell us the actual cards. TDP from the manufacturers are not really what the card actually uses. We would also need to know how many hours a day you would be gaming and how many hours a day the computer would be off entirely to be accurate.
The 6870 and 6950 use the same power at idle so we can take that out of the equation. The 6950 only uses about 15w more than the 6870 under load. You seem to be saying about 25 hours of gaming a week. Over the course of a year the difference on your power bill using the national average for energy costs would only be about $4.
Assuming you play games 35 hours per week, every week, that works out to a total of 72.8KWH difference in one year. If you are paying $0.1 per KWH then that works out to $7.28. If it's $0.20 per KWH, then it would be $14.56.
here are two sample pages: http://www.guru3d.com/article/his-radeon-6970-iceq-turb... http://www.guru3d.com/article/his-radeon-6870-iceq-x-tu...
pay close attention to how the card power consumption is calculated at different websites. some sites use card power consumption, some use total system power consumption at the wall socket, some use power consumption delta using another card as reference. most review sites turn off power saving and protective measures to measure maximum power consumption.
imo, the 6870 has lower tdp and lower load power consumption. so if the rest of the system specs are identical, the 6870 will use less power.
it's good to be considerate to your parents and power consumption.