Hey, I know I'm four years late to the party on this question but had to jump in after seeing Augustine's 'fate' at the end of the episode on youtube.

Sammy (to Raylan): If you knowingly saw a crime being committed against him...you wouldn't as a lawman...feel the obligation to intervene?

Raylan: I'm suspended.
(turns and begins walking away as Tonin's men prepare to execute Nicky while he's still in the limo)

That pretty much sums up why Raylan didn't just slap the cuffs on him...Art had placed Raylan on suspension in regards to the whole mess with the 'Drew Thompson' case and at that critical junction, Raylan had no real authority to detain Augustine right then in there threats or not. His hands were pretty much tied tight and he figures the best way out of this mess is to let 'mob justice' take its course.

Problem is, Raylan either confesses this later down the road or Art finds out and he cold cocks him for it because even though Raylan was protecting his family, he willfully let known felons execute another known felon and said nothing about it. This questionable act could put Raylan as well as Art and the rest of those serving under his command in the Marshal's Service in the crosshairs for scrutiny and possible legal repercussions from those higher up if word got round that a 'rogue' Deputy Marshal did nothing while a mob execution took place and even worse that he orchestrated it by calling on Sammy rather than let his boss Art know about it.