The character has changed his name, but not his appearance
In Phandalin Locations, the Quest Finding Iarno says:
Unknown to Sildar, Iarno created the Redbrands, installed himself as their leader, and took the alias Glasstaff to conceal his identity.
and
Sildar describes Iarno as “a short, dark-bearded human wizard in his thirties.”
...
While the flavor text for the party's encounter with the character says:
Sitting at the desk is a short, dark-bearded human male in robes, studying a tome. [Redbrand Hideout - Area 12 (Glasstaff's Quarters)]
Sildar's description of the character exactly matches the description of the character when the PCs meet him. It is clear that the character has made no effort to change his appearance, but only his name.
In Sildar's Background, we learn:
After questioning several locals, Sildar learns that Iarno Albrek, a fellow member of the Lords’ Alliance, disappeared while exploring the area around Tresendar Manor about two months ago, shortly after arriving in Phandalin. Sildar asks the characters to investigate the manor and the surrounding area to find and bring back Iarno—or what’s left of him, if something killed him.
So we know that the character passed briefly through the town and then 'disappeared'. It has not been necessary for him to change his appearance since no one in town has seen him since.
Putting all of this together, Sildar and the character last saw each other a few months ago, and neither of them have changed their appearance since, although the character has taken on a new name. They should both immediately recognize the other on sight.
What happens next?
The adventure is quite clear on how Sildar will react:
In Phandalin Locations / Quest - Finding Iarno:
Once he learns the truth about Iarno, Sildar expresses a desire to have the wizard captured and transported to Neverwinter to face the judgment of a higher authority.
...
While in Overview of Act 2: Redbrand Hideout : Developments:
If Iarno is taken into custody, Sildar Hallwinter arranges to have the wizard incarcerated in the townmaster’s hall until he can be safely transported back to Neverwinter.
So we know very well how Sildar will react to this discovery.
The character's reaction, however, will depend on the situation.
We are told that Iarno is evil and that he wants to "line his pockets" and "secure his own position in town". We are also told that no other member of the Lord's Alliance knows his true identity, so he would likely want to keep things this way. If he wins a fight with Sildar present, he might offer Sildar a chance to join him, but he will not let him report back to the Lord's Alliance and that might mean imprisoning or killing him.
If he loses a fight, we are told that:
If he is reduced to 8 or fewer hit points and has no avenues of escape, Iarno surrenders. He values his life more than anything, and he remains a model prisoner in the hopes that the Black Spider will somehow learn of his predicament and “arrange for his freedom.”
In fact, if Sildar is not present when the party defeats Iarno, he might even reveal his identify to them and remind them that Sildar will want him alive.