Aw is an interjection used when someone thinks what they see is adorable, or when he or she is disappointed. Some dictionaries list a variant spelling if aww, but this is vastly less popular than the one w spelling.
Awe is a noun for the feeling of wonderment after seeing something spectacular. One can also be in awe if he or she is afraid of someone or respects someone greatly. As a verb it means to cause someone else to be in awe.
The adjective awe-inspiring uses the latter of the two words. Though technically something could be aw-inspiring, most would see that term as a misspelling of the more popular version.
In short, when using it as a sound or interjection, spell it aw. As a noun or verb, add the e.
Examples
She did, however, take to her Instagram to post a picture of a framed picture of her kissing her new fiance with the caption, ‘Love’. Aw, how sweet! [Hollywood Life]
Officials have maintained, when it comes to taking any real action, their hands are tied by bylaws that don’t provide an immediate avenue for removal. Aw, too bad, right? [Detroit Free Press]
Now 10, Lacy watches in awe the video of her mom and dad’s wedding day on July 27, 2002, when more than 1,000 people overflowed the pews at First Baptist Church in Bay City, Texas. [CNN]
The blood moon is a natural awe-inspiring spectacle to most, but to some this series of blood moons signify judgment day along with signs seen in the Bible as warnings. [Examiner]