I care a lot about the aesthetics of a game, but for me it's not so much graphical power as it is aesthetic execution.
Let's put it this way: visually speaking, Combat Evolved is my favorite game of the Halo series. Yes, its graphics are old and extremely low-powered, but within the engine the artistic execution is brilliant, particularly in terms of lighting and color schemes. The clean, crisp look is IMO much nicer on the eyes than the highly cluttered look of Halo 3.
On top of all this, it doesn't hurt that the graphical integrity is extremely good; Halo 3 suffers from horrible draw distances on many kinds of objects to boost performance, and Halo 2 does a lot of very obvious texture popping, and all the games after CE deload things fairly quickly. Contrast this with CE, where all that goes on is extremely minimal texture popping, helping a lot towards making the game feel "solid."
So yeah, that's pretty much my take. Graphics matter a LOT, but within that realm I care a heck of a lot more about artistic execution than graphical power.
edit: Of course, my answer might seem kind of funny, since "good graphics" is of course usually taken to be synonymous with "powerful graphics", which always grated too hard with my intuition's use of English to fully accept.