The platoon splits on different pitch types are well documented: John Walsh calculated them in the 2008 THT Annual, I did in these pages, and recently Max Marchi broke down the pitch types into finer buckets and showed the splits for each bucket. Here I am interested in understanding, at least in a qualitative sense, why different pitch types have different platoon splits. In no way is this going to be a complete explanation, but an attempt at a first step. Here I am going to focus on the slider, a pitch with a large platoon split (much better against same-handed batters), and the changeup, a pitch with no platoon split (does roughly the same against same- and opposite-handed batters).

Since almost all pitchers pitch off of the fastball I think it is best to compare both pitches against the fastball. Here is a chart I made for the 2009 THT Annual showing the approximate movement of the different pitch types for right-handed pitchers.

Using the four-seam fastball as a guide you can see that a slider in comparison moves down and away from same-handed batters (in to opposite-handed batters). The changeup moves moves down and in to same-handed batters (away from opposite-handed batters). I think this is part of the reason platoon split for the two pitches.

If a pitcher can release his fastball and slider with roughly the same initial trajectory and locate his fastball around the middle of the zone the difference in movement will put his slider down and away to a same-handed batter and down-and-in to an opposite handed batter. If he does the same with his changeup the pitch will end up down and away to the opposite-handed batter and down and in to the same-handed batter. All else being equal a down-and-away pitch is much better than a down-and-in pitch. Looking back at my run value by location maps down and away is the best place to pitch, while down and in is, other than the heart of the strike zone, the worse place to be in within the zone.

So when a pitcher repeats his motion well with his pitches, starts his pitches on roughly the same trajectory and locates his fastball in the zone the movement relative to fastball movement will take a changeup into a good spot against opposite-handed batters and a poor spot against same-handed batters and vice versa for a slider. This, I think, is a big reason for the platoon splits, or lack thereof, for the two pitch types.

Another source for the platoon split is the different vantage points a batter has against same- and opposite-handed pitchers. A same-handed batter most likely doesn't get as good a view of the pitch as it is released. Mike Fast takes this into account very well by showing pitch trajectories from the view point of the batter (Scroll a little more than half way down this post to see).

Josh Kalk theorized that minimizing the difference between a slider and a fastball along the beginning of their trajectories might be a key to a slider's effectiveness. I thought it would be cool to check that out from a same-handed versus opposite-handed batter's perspective. My physics chops are not the equal of Mike Fast's so I sort of fudged the perspective projection.
From RHB's perspective:

From LHB's perspective:

These are a subsequent fastball (red) and slider (blue) from Brad Lidge, a prototypical fastball-slider pitcher. The black dots indicate the pitch location 0.075 seconds into the pitch's trajectory, approximately when a batter must decided to swing or not. To the right-handed batter the two trajectories are almost identical up to that point. The fastball is slightly farther along but it is very close. For the left-handed batter the two pitches appear much farther apart. My perspectives are not perfect, but I think this could indicate another possible reason for sliders' large platoon split.