These two cards are close enough in raw performance that FPS shouldn't be what decides this purchase. Power draw should.
The performance gap barely exists
Used, the 7800 XT runs $79.99 more than the 6800 XT on ChipFlip's tracked listings, a 20% premium. TechRadar's own testing found the two essentially tied at 1440p without ray tracing, with the 7800 XT only pulling a few percent ahead. The 6800 XT actually posts a slightly higher aggregate benchmark score in some comparisons. Neither card is meaningfully faster than the other for most games at most settings.
What actually separates them
The 7800 XT draws 263W against the 6800 XT's 300W, a real, ongoing difference that adds up over years of use, not just a spec-sheet footnote. It also runs on a newer, more efficient manufacturing process and includes AI accelerator hardware the 6800 XT simply doesn't have, useful for generative image workloads the older card can't touch.
When the 6800 XT is the smarter buy anyway
If it's priced meaningfully below the 7800 XT, take it without hesitation. Near-identical gaming performance for less money is a straightforward win, and the power difference only matters if electricity costs or heat output are a real concern in your setup.
The takeaway
Don't pay the 7800 XT's premium expecting a real performance jump, because there mostly isn't one. Pay it for the lower power draw and the AI acceleration hardware if either of those genuinely matters to you. If neither does, the cheaper 6800 XT is the better dollar-for-dollar pick.