Two right hands share this instrument and they have almost nothing in common. One strikes downward with the back of a fingernail; the other wears picks and rolls upward through the strings. Both are called banjo playing, both use the same fretted shapes, and a beginner who does not know the fork exists can spend six months learning the wrong one for the music they actually like.
Clawhammer
The hand stays loosely closed and moves as a unit. The index or middle fingernail strikes down onto a melody string, the hand keeps travelling to brush the others, and the thumb catches the fifth string on the way back up. That is the bum-ditty: a note, a brush, a drone, endlessly. The drone is not decoration here — it is structural, and it is why clawhammer sounds ancient and rhythmic rather than florid.
What follows from it: clawhammer lives comfortably in alternate tunings, because the tune is being picked out of the open strings as much as fretted, and retuning changes which notes are free. Old-time repertoire, fiddle tunes played at a steady dance tempo, and a lot of singing accompaniment.
Bluegrass three-finger
Thumb, index and middle wear picks and move independently, upward through the strings, in the fixed patterns this site calls rolls. The stream of eighth notes is continuous and the melody is implied by which notes inside the roll the left hand has stopped. It sounds busy because it is: eight notes to the bar, every bar.
What follows from it: bluegrass mostly stays in open G and moves key with a capo rather than a retune, because the rolls are built around specific strings and changing what those strings are would break the pattern. Bluegrass repertoire, breaks over a band, and backup that keeps time for everyone else.
What actually differs
Direction of attack, and what the fifth string is for. Clawhammer strikes down and uses the drone as a rhythmic answer; three-finger picks up and uses the drone as a pitch inside a pattern. Everything else people list — picks, open-back versus resonator, tunings, repertoire — follows from those two.
What does not differ
The left hand. Every chord shape in the chord library and every note in the scale maps is identical in both styles, because both are the same instrument tuned the same way. If you switch later, none of that is wasted.
Choosing
Play recordings of both and pick the sound, not the reputation. Neither is a beginner version of the other, neither is harder in any general sense, and a great many players end up doing both. If you have not bought an instrument yet, the choice does bear on it — the first-banjo guide explains how.