Adam lacks in confidence because his family don't consider him good enough a lawyer to be part of the family firm. In addition, he is somewhat overwhelmed by the scale of the trial and media attention, along with the seeming mountain of evidence against his client. When Gus and Shawn first see him, he's accidentally alienating the jury by his unsure delivery and unfashionable dress. With the dynamic duo's help, however, he gets a rapid makeover, and is led to the vital information which will exonerate Sandra and put his name, like that of so many Hornstocks before him, on the company's headed paper—as a partner. Now full of confidence, Hornstock feels emboldened to ask out Juliet (to Shawn's dismay).
He and prosecutor Logan Phelps seem to have a healthy rivalry going on.
In the original episode, Adam's final interaction with Shawn isn't about whether he can ask out Juliet, but rather about whether he can help Shawn buy back his motorcycle from the impound auction.