The ticket icon in Bear School denotes a bear that can help you meet the unlock requirements for one or more Tiers of a Special Event. Thus, you may want to level up that bear even though he/she may not otherwise seem particularly useful to you.
In the example screenshot above, Superstar Bear has a ticket icon because she is a "celebrity" bear, and there is a Special Event Tier—"Walk of Fame," Tier 2—that can only be played if you own 3 or more "celebrity" bears of level 20 or higher. Thus, leveling her up can help you to eventually unlock that particular Special Event Tier. Victrola Bear and Anubis Bear do not have ticket icons because leveling them up doesn't help you to unlock any Special Event Tiers.
One nuance to this: Bears will not display ticket icons until you meet the requirements for playing Tier 1 of the relevant Special Event. This is an intentional design decision—we don't want players to be overwhelmed with too many ticket icons, so we initially hide the ticket icons that won't be relevant until a ways in the future.