I level up, add my stat points (strength, agility, etc.), and when I load again, my level is higher, but my stats are all at the default levels. This happened with my GOG Version of the game and the steam version. I don't get what the problem is. I added five points to agility and nothing, and when I reload, the points are gone.

Are you playing on a netbook? There is a checkmark button at the bottom of the stats window to confirm your distributed points, but at lower screen resolution it might not be visible on screen. As you found, saving and reloading with distributed but unapplied stat points will result in their loss.

If you don't have a save you can revert to before distributing and loosing the stat points, you can use iZakaroN's SaveEditor (or from Kiya's site, with description) to get them back.
Download the editor and extract the zip file somewhere, then run the DDSaveEdit.exe program. Next, click the '...' button at the top right (by the 'Save File' text box), browse to the DD savegames folder, and then in your latest save select the data.000 file and open it. Next, click on 'Redistribute Points', then in the 'Free Points' box just above there, type in the number of stat points you should have and then click 'Save'. Once you re-load that save game your stat points will be back, and you can distribute them.