The Meshech Chochmah has a novel approach to these two words. He writes that אל has more of a tone of a request (although still obligatory). That's why נא, please, only ever accompanies אל (such as Genesis 18:3), and never לא, which is more of a more forceful command.
He proceeds to give several cases in Chumash where this proves to be true. Rav Kupperman, in his notes to Meshech Chochmah, points out that these examples show three types of cases where the Torah will choose אל instead of לא:
When the desire of the Torah is to command something, but for one reason or another Hashem didn't want to make it a full-forced command. He sees this from Leviticus 25:36, which sounds like it's prohibiting lending to a ger toshav with interest (which it isn't, as it's only assur miderabannan, see Bava Metziah 70b. He's saying Chazal saw the need to prohibit from this pasuk which uses אל).
When the prohibition is so self-evident it doesn't need to be written in a strong form, like Leviticus 25:14, the prohibition to overcharge another (he says from societal sense it's a bad idea).
Sometimes it's simply not feasible to have a mitzvah stated in the form of a definite command. An example is Leviticus 19:4, which makes a prohibition on a person's thoughts, which are hard to control.