Aww, aw, or awww, as in: Aww (aw or awww), that's so cute. Aww (aw or awww), that is amazing. Aww (aw or awww), she's got the best guy ever. Aww (aw or awww), congratulations on your promotion.
Awe, as in: Wow I'm in awe over this magnificent scenery. She was in awe over just how much money had been raised. The view was awe inspiring.
Aww, aw and awww are all versions of slang. Awe on the other hand is an actual word and should be used as such.
Awe: A mixed emotion of reverence, respect, dread, and wonder inspired by authority, genius, great beauty, sublimity, or might: We felt awe when contemplating the works of Bach. The observers were in awe of the destructive power of the new weapon.