Tartaric acid is the main acid in grapes. Usually, this precipitates out when juice is concentrated, so kit manufacturers add more back in, but usually, they add a blend to achieve the acid profile they want for the finished product. This usually means more malic and citric, especially in whites, but also in reds.
So, the more concentrate used = the less concentration of tartaric acid. Of course, this also directly correlates to ripeness of grapes too. The acids in grapes undergo a change as the grape gets riper too, causing more tartaric to malic concentrations. The conclusion is that the better and closer the grape is to ripe, the more tartaric will be in there. Of course, this is all varietal specific too, as some varietals when ripe, start to lose a lot of acid as well (Syrah in particular, as well as Merlot). However, the higher the quality of the juice/grapes they came from (crop yield, ripeness, amount of leaf cover, sunburn, etc) the more likely the case of bitartrates falling out of solution.
And no, it doesn't mean that higher end kits have more acid, but they usually have a greater concentration of tartaric acid from natural sources, versus a blend of acids needed to be added later due to higher amounts of concentrate.