If you have NIghtingale Armor but aren't wearing it, and you encounter a Thief in the wilderness, the Thief will not recognize you as a Guild member or a Nightingale, but you should get the dialogue option: "I would think you'd have more respect for a Nightingale." At which point he/she will leave you alone.
If you're wearing the Nightingale Armor the Thief should recognize you as a Nightingale and will give you 500 gold and then be on his way.
Here's the thing; I finished the entire Nightingale Arc and, while wearing the armor, Thieves don't recongnize me as Nightingale nor do I have the dialogue option.
Have you tried exiting the conversation, using an illusion spell to calm him down (or the racial ability of imperials, if you happen to be imperial) and initiating conversation again? I have a similar problem with dialogue options not appearing when talking to followers, which I can fix by exiting conversation and talking to them again.
About other characters, mostly guards will comment on what armor you're wearing, vigilants of stendarr will attack if you're wearing a full set of daedric armor, and I'm sure there are some other special events that happen if you're wearing a specific type of armor.
Thanks Che'Darr, I'll try that and get back to you.
About guards commenting; they don't comment on my Nightingale Armor either. They should say something like, "That armor, it's not like anything I've ever seen... Like forged Midnight." but they never, ever have, not in all the years I've had this game xD