What's the difference between "darf" and "kann"?

In English you may hear people say "Can I" to ask for permission from someone to do something. But this is a common grammatical mistake that even native speakers of English make.

"Can" is about the ability to do something while "may" is about the permission to do that.

One can kiss anyone as long as they are physically capable of doing so. However one may not kiss anyone. They have to have the other party's consent.

In German it is similar.

Kann corresponds to Can
Darf corresponds to May

können refers to ability or possibility. dürfen refers to permission by another person. But in colloquial language können is often used instead of dürfen, especially when talking to persons of equal standing.

"Darf ich" means "am I allowed to" or "may I", while "kann ich" means "can I".

Same as in English, the expression "kann ich" is often also used when asking for permission, but the main difference between the two is, that "kann ich" means to have the ability to do something, while "darf ich" only means to have the permission to do it.

Another example would be:

Darf ich nach Hause gehen? --> May I go home?

Kann ich nach Hause gehen? --> Can I go home?

Of course they both ask for permission in the context, but literally you can also do it without having permission. So literally you can although you may not, so "du kannst, obwohl du nicht darfst". Usually they are used interchangably in such questions, same as "may" and "can".

"Kann ich" is generally more informal, and "darf ich" asks more directly for one's personal permission, so in your example "Darf ich dich küssen" sounds more appropriate to me.

Kann ich dich küssen?

Is translated to:

Am I able to kiss you?

While

Darf ich dich küssen?

is translated to:

Do I have the permission to kiss you?