What's the difference between αγαπώ and αγαπάω?
Same thing. This type of verbs (ending both at ώ and -άω/-έω/-όω) are called συνηρημένα. Other examples: τιμάω-τιμώ (honor), ποιέω-ποιώ (do, create, make). Generally you can use them interchangeably.
Is that a regional thing (i.e. -αω vs -ω usage varies in different parts of Greece)? If not, from where does this difference arise?
Ι wouldn't say so. In fact, as a native speaker, I'm pretty sure it's the exact opposite that's happening. :P Either way, both are correct ^.^
Isn't it the same as in English? "Love" and "like" can both be used for food (though not recommended by everybody). It didn't accept "like."
While in a conversation we can use whichever word expresses what we feel when we are translating we need to use the word the represents that used in the source language. Here the Greek says "love".
Oh, yeah, sorry about my lapsus linguae in comments. I think I did put “I like to eat chocolate”, and it got taken down as incorrect. Someone else on this thread stated that too
Well, I do think that's because αγαπάω (love) is a much stronger word than αρέσω (like). Το be honest, this structure is not something you'll come across too often in speech ("Αγαπάω/Λατρεύω την σοκολάτα" is probably more common.), but for teaching purposes, it's better not to mix different verbs up. ^.^
This reminds me of the book by C.S.Lewis, "The Four Loves." Four different types of "love."
Would the translation "I love eating chocolate" be valid, or would that be a different sentence structure?
It's the same because Greek doesn't have a present continuous tense. (And btw way me too...love to eat and eating chocolate )