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In ROTK, Chapter 2 of Book Five, "The Passing of the Grey Company," Aragorn tells Gimli and Legolas that he has looked into the Orthanc stone and had a chat with Sauron. He says that Sauron "saw me, but in other guise than you see me here." What was this "guise"? And why would Aragorn appear in this fashion when he also revealed himself to Sauron as Isildur's heir?

I think the answer to your question can be found in the context of the words you quoted:

'You have looked in that accursed stone of wizardry!' exclaimed Gimli with fear and astonishment in his face. 'Did you say aught to - him? Even Gandalf feared that encounter.'

'You forget to whom you speak,' said Aragorn sternly, and his eyes glinted. 'Did I not openly proclaim my title before the doors of Edoras? What do you fear that I should say to him? Nay, Gimli,' he said in a softer voice, and the grimness left his face, and he looked like one who has laboured in sleepless pain for many nights. 'Nay, my friends, I am the lawful master of the Stone, and I had both the right and the strength to use it, or so I judged. The right cannot be doubted. The strength was enough - barely.'

He drew a deep breath. 'It was a bitter struggle, and the weariness is slow to pass. I spoke no word to him, and in the end I wrenched the Stone to my own will. That alone he will find hard to endure. And he beheld me. Yes, Master Gimli, he saw me, but in other guise than you see me here. If that will aid him, then I have done ill. But I do not think so. To know that I lived and walked the earth was a blow to his heart, I deem; for he knew it not till now. The eyes in Orthanc did not see through the armour of Théoden; but Sauron has not forgotten Isildur and the sword of Elendil. Now in the very hour of his great designs the heir of Isildur and the Sword are revealed; for l showed the blade re-forged to him. He is not so mighty yet that he is above fear; nay, doubt ever gnaws him.'

and much later, Gandalf says:

' As Aragorn has begun, so we must go on. We must push Sauron to his last throw. We must call out his hidden strength, so that he shall empty his land. We must march out to meet him at once. We must make ourselves the bait, though his jaws should close on us. He will take that bait, in hope and in greed, for he will think that in such rashness he sees the pride of the new Ringlord..."

(My emphasis.)

He must have shown himself as Isldur's Heir. And Sauron had good reason to fear Isildur's Heir: Isildur struck off his hand when Sauron still had the Ring. This heir could control the power (waning though it was) of Gondor. And this heir was a man of great power -- he went mano-a-mano (so to speak) against Sauron and won.

So Sauron knew that the Ring was abroad, that the Heir of Isildur lives, was not a weakling, but could challenge him to a duel of wills and win -- and might soon have the Ring. By appearing as Isildur's Heir, Aragon showed Sauron his worst nightmare.