Mash thickness affects mash pH (less water means less alkalinity for the grains to counter). Mash pH can impact efficiency, mouthfeel and taste of the final beer. Thus, mash thickness has an impact on the final beer, but the impact is probably quite minor compared to other factors.
pH also affects the enzymes (alpha- and beta-amylase favour different pH's) so in theory mash thickness could affect attenuation. In practice, this is not measurable at a homebrew level (i.e. effectively zero).