Which of these 2 ATI is better for my system?

For sure :)

Thanks guys for the research, much appreciated.

I understand that 7770 actually uses less power (80 W?) than 6770? Then it really is a good upgrade in that way also, seeing that it also lowers my power bills while upping the performance (even if by a small margin, I guess). I've been also hearing that the new HD 7xxx series uses a new technology that offers better power management and gaming power?

Go as high as your budget allows OR not buy anything if thinking the 7770, since it is not a good jump for the cost, unless your 6770 is busted, I'd go higher....

Power wise I would ignore the increase in watts, unless your electricity per kilowatt is incredibly high, you probably won't even notice it.

.....still dragging your feet, eh?

Try this:

AMD Radeon HD 7790 Video Card Review w/ Gigabyte & Sapphire - Power Consumption - Legit Reviews

Looks like around 100w is what a typical 7790 will be when idle.

If you're a gamer and want performance, you have been warned to go 7850 or better! More than once!

Heh , nah it's okay.
It's a matter of cost-performance ratio mainly. Lemme see what the cards you suggested cost on the Indian market. If within my budget, I'll go for it. Or if not I have to opt for HD7770 (my r6770 is showing definite signs of age), as of now I'm in for investment in a Logitech G400 or suchlike, as my trusted old Abyssus is now deceased. Thank you for all the research and time :).