While discussing funding for PBS, one of the aides talks about licensing for the Muppets and mentions Fuzzy Bear which Toby corrects as Fozzie Bear. Fozzie Bear is not a Muppet featured on Sesame Street but is part of the cast of The Muppet Show and the various movies. Kermit the Frog, Rowlf the Dog and a few minor characters are the only muppets to appear on both Sesame Street and The Muppet Show.
The goof item below may give away important plot points.
In talking with the aide who leaked his rehab stint, Leo says he hasn't had a drink in 6.5 years. Yet, in Season 3's "Bartlet for America," we see Leo go on a drinking binge before a debate during Barlet's first presidential campaign, a period just two years before this meeting.