(Spoilers for both, natch.)
The thing that made me realize this was the fact that both of their mothers are captured by the natives of their desert planet. Paul saves his mother's life and befriends them; Anakin fails to save his mother and kills them.
Then you have both of them involved with a royal woman. Paul marries Irulan for show but privately is distant from her. Anakin marries Padme privately but in public keeps it a secret.
Anakin is considered the Chosen One by a magical order, who test him and try to train him. Paul is tested but assumed not to be the Chosen One, and the magical group writes him off. Which is perhaps the smart way to go, given that Anakin destroys the group that tries to help him while Paul leaves his pretty much alone after showing them who's boss.
Speaking of boss, Anakin creates an Emperor, Paul overthrows one. Anakin offers his son rulership of the galaxy but is rejected, Paul successfully passes it down (although here Luke might be considered luckier than Leto II).
Also Anakin, maimed after having been considered dead for some time, redeems himself, saves his son and changes the course of history (by killing the Emperor); Paul, blinded and considered dead for some time can do little more than tell his son how hard it is to have any good effect on history (although here my memory is hazy).
And to sum up: Anakin hates sand. Paul cherishes desert power. Desert POWER.
Would welcome further parallels or exceptions!
Addendum: Paul's imperial troops wear robes, are armed with knives, and are mega-badass. Anakin's wear armor, are armed with lasers, and are useless.