As mentioned in the comments, Light, Medium, and Heavy don't actually correspond to weight (that would just make too much sense, wouldn't it?). Rather, they describe how good they are at what their specialty is.
That is, Light will give the least amount of boost, Medium will give more, but Heavy will give the most. There are no drawbacks from using Heavy over Light (even for the stealth armor...).
The posts in this thread describes it quite well:
The heavier they are, the better they are at what they do. Armor weight doesn't actually affect you. It's just a straight upgrade to their particular stats.
For example, the silent hunter heavy armor improves your stealth more than the medium, which improves it more than the light.
Edit: heavier armors usually have more mod slots than their light counterparts as well.
Also, from a cosmetic standpoint, the heavier versions usually have a few extra pieces attached.
A more detailed guide on what armor ranks/stats mean can be found here if you want additional information.