The Thrown Dagger proficiency gives the character proficiency in any knife or dagger designed to be thrown, as well as the ability to throw two such knives per round. These daggers are usually thrown simultaneously, and adjudicated by rolling two dice. Both throwing daggers and throwing knives are considered to be the same weapon, although stylistically they vary greatly. For game purposes, proficiency in one is proficiency in both.
The proficiency allows the character to wield a thrown dagger or knife as a melee weapon, although these blades usually do less damage (typical throwing daggers do 1d3/1d2 as melee weapons.)
The Thrown Dagger proficiency gives a "related weapon" penalty to most small blades that aren't thrown.
The Dagger proficiency, on the other hand, gives the character proficiency in any dagger or dagger-like weapon, including the thrown dagger when wielded as a melee weapon, and gives a related weapon penalty to small blades (such as the throwing knife) when thrown. The dagger proficiency includes all piercing small blades that are wielded in similar ways, including dirk, kris, and stiletto. Main-Gauche, Sai and Parrying dagger may be wielded without penalty by those proficient in dagger, but you can't get the AC bonus from them when wielded that way (essentially you're using them incorrectly.) Jambiya, (slashing) knives and Katar/Punching dagger, on the other hand, are their own proficiencies, requiring specialized training. These "small blades" take a related weapons bonus.
The Knife proficiency gives the character in slashing small blades, including the knife, folding knife, and, counter-intuitively, some daggers designed to slash, not pierce. This is made more complex by the fact that some weapons in the dagger class are considered piercing/slashing, and can sometimes be wielded with either proficiency.
The DM has a lot of leeway in deciding if a weapon takes a minus in these departments, and one way we reduced the 'stress' of having to pick in our own game was increasing the base weapon proficiencies for all characters by one slot, allowing even a mage to take both knife and dagger if the thought of not being able to wield them all was such a problem.
The chart below shows some of the common knives/daggers and how we place them on the proficiency chart: