In a small aircraft, the turn indicator (and the slightly better turn coordinator) is usually electrically powered, while the attitude indicator is vacuum powered. If the attitude indicator fails, either mechanically or do to vacuum fail, the turn indicator can assist with keeping the wings level in instrument conditions.
The slip indicator is merely a ball in a fluid and is not subject to any electric or vacuum failure, and can be a backup for an electronic flight display's slip-skid indicator. In attempting a coordinated turn, the ball can be "stepped on" via the rudder to prevent undesirable yaw. In some aircraft, rudder isn't necessary in a turn or initial bank as yaw is negligible.
Under Instrument Meteorological Conditions (IMC), a skid at or near stall speed indicates a spin (usually undesirable), and the instrument can assist with recovery. A slip at speeds well above stall with rapidly increasing airspeed indicates a spiral. The spiral may be due to misleading attitude indicator, and the indicator and ball can assist in recovery from a spiral.
In a multi-engine aircraft under instrument conditions, yaw (and heading change) is one of the first indications of an engine failure of some sort due to asymmetric thrust, and "stepping on the ball" and raising the wing of the dead engine can restore stable flight.