Sechitah: Squeezing liquid out of a solid in which it was absorbed is prohibited on Shabbos. When wet, densely packed nylon bristles of a regular toothbrush are pressed against the teeth, many poskim consider it is as if the water is squeezed out of the bristles and forbid using a wet toothbrush. The rubber bristles of the Shabbos Toothbrush™ are far enough apart that it is obvious that liquid does not get absorbed into the bristles and is not squeezed out Shabbos Toothbrush's soft rubber "bristles"
Chavalah: Causing oneself to bleed on Shabbos is prohibited. Sharp nylon bristles of a regular toothbrush can cause gums to bleed. The soft rounded rubber bristles of the Shabbos Toothbrush™ do not cause bleeding. In the extremely rare case that very sensitive gums bleed even with the Shabbos Toothbrush™, consult your local Rabbi and see your dentist!
Memareach: Spreading a cream or ointment onto an object used as an applicator to the body is prohibited on Shabbos. Therefore, most poskim prohibit use of regular toothpaste or gel applied to any toothbrush. Kosher Innovations™ Shabbos Toothwash™ is a concentrated liquid and this prohibition does not apply. We do not recommend using the Shabbos Toothbrush™ with any product other than Kosher Innovations Shabbos Toothwash™ Also, the Shabbos Toothwash™ is certified kosher by the OU.
Uvdin D'Chol: If a regular activity involves something that one may not do on Shabbos, that activity will be classified as a “weekday activity”. For some poskim who permit the use of a regular toothbrush without toothpaste this prohibition would apply to using one's weekday toothbrush and they require a separate and even distinctive toothbrush for Shabbos. The Kosher Innovations Shabbos Toothbrush™ was designed specifically for Shabbos and not for weekday use, eliminating the issue of Uvdin D'Chol according to all poskim.
According to a list found here, Sechitah (whether a Tolda of Dush or Melabain) and Memachaik (a rabbinic prohibition based on Memareach) are forbidden on Yom Tov, while Chavalah (a Tolda of Shochet) and is permitted.
If one has a unique toothbrush set aside for Yom Tov (to avoid an Uvda D'Chol) and does neither wet the toothbrush (to avoid Sechitah) nor apply toothpaste (to avoid Memareach), it would seem that the sole act of brushing the teeth with a standard toothbrush (even if it causes bleeding) would be permitted on Yom Tov.
However, I have never seen or heard this stated explicitly. Neither has R' Yakov Farkas, a Wiznitzer Dayan in Montreal, but he could also not think of a reason it would be prohibited.
* They mistakenly call Memareach and Uvda D'Chol – which are rabbinic prohibitions – "categories of prohibited melacha".