I've never really understood the difference between the two. How does the game make checks for each one? How high should I raise each one?

As I understand it you need to have high scores in both.

The first one is used when you initially hide in shadows. That determines if you hide sucessfully. if you have a low score here you may see a "Hide in Shadows Failed* message when you try to hide.

The Move silently is used when you are moving around. When you are hidden and moving this is constanly checked to see that you remain hidden.

Hopefully I got this right as its been a while.

<span style="color: lightblue">Chance to Hide = (HiS + MS) /2 . This value is checked when you try to hide, and - I believe - once each round you continue to hide. You want this value as high as possible.
Move silently gives bonuses after you leave the shadows. In particular, it increases the delay between when you leave the shadows and when you become visible.
Hide in Shadows does nothing extra for you.
So, forget that Hide in Shadows exists. Put all your points into Move Silently.

Hoooowever, there is a numerical limit on how high you can pump your Move Silently, which is 255%. So if you wish to have a higher Stealth value( say 150%), then some points will be needed in Hide in Shadows.

Pump into Move Silently first though.

Wow, this is so cool (never knew this!). I am just bookmarking this thread for myself!