"I feel like Bill Murray in 'Groundhog Day' because a year and a half (ago) we were talking about another interesting social media company with a big user base and tremendous potential in online advertising," Damodaran said. "Now we're talking about Twitter. I'd wager in six months we're going to be talking about Pinterest and Snapchat, using exactly the same terms."
Damodaran, who doesn't own shares of Twitter, said that fair valuation was around $18 and that he might be a buyer in the $12 to $15 range.
(Read more: 'You'll have other opportunities' in Twitter: Josh Brown)
But a bull on Twitter stock said there was plenty of upside for the stock.
"We're at a 300 percent revenue growth rate," said Victor Anthony of Topeka Capital, adding that when figuring a valuation for a hyper-growth company, it was important to adjust multiples for growth.
"The stock is actually trading in line with other Internet media names, as well as other media technology stocks," said Anthony, who does not own Twitter stock. "I actually see more upside in the stock over the next year."
Anthony, who has a $54 price target and a "buy" rating on the stock, said Twitter is developing its own ecosystem.
(Read more: Wait for a better price to buy Facebook stock: Pros)
"I think Twitter will, over time, become a significant Web platform similar to what you see with Google," he said.
As for the comparison to Facebook's initial public offering, Anthony said it wasn't exactly comparable.
"When Facebook IPO'd, there were two secular trends that were working against them. One is that social media advertising hadn't been proven, and second, mobile advertising really hadn't been proven. Twitter had a hard time monetizing mobile," he said. "I think the runway for them is both long and wide for the next several years."
— By CNBC's Bruno J. Navarro. Follow him on Twitter