difference between crystals in applications
There are indeed MCUs that will use a 32.768 kHz crystal to run their primary oscillator that then can get stepped up to a much higher frequency using a PLL for the main CPU clock. The primary reason for operating in this manner comes from two usage scenarios: 

The first one is where the MCU has an built in RTC that uses the fundamental 32.768 kHz to run the RTC (real time clock). 

The second scenario is where the MCU is operating on a battery and to extend battery operation as much as possible the MCU is placed into a sleep mode most of the time. In sleep mode the PLL is shut off and the MCU chip is left operating at 32.768 kHz which consumes much less power than when the PLL is active running at many MHz. 

32.768 kHz crystals will come in various types and specifications depending on the intended application. Some factors to consider:


Crystals are specified to operate within a frequency band of their nominal specified frequency. An error specification in ppm (parts per million) will give an indication of how accurate the frequency will be in normal operating conditions. Expect to pay more for lower ppm numbers. 
Crystals are typically specified to work with a specific load capacitance on the component. The part's frequency, stability and tolerance specifications are based upon this specific capacitance value. Sometimes it becomes important to evaluate the stray capacitance of the PC board traces and input pin capacitance to the MCU to select the crystal suitable to the application. It may even be necessary to compensate by adding additional capacitance to the circuit. 
Long term life of a crystal in terms of its frequency accuracy can be dependent upon the drive strength of the signal from the oscillator. In RTC applications this is intentionally minimized as much as possible to ensure long term accuracy of the clock. In a pure MCU application this may be much less of a concern and a different type of oscillator circuit may be used. More drive ensures faster oscillator startup time. Less drive can make it easier for the oscillator to stop running due to stray influences. (Some RTCs can stop operating simply by touching the crystal circuit with your finger). The construction and characteristics of a specific crystal can be optimized for lower drive versus higher drive levels but at the same time expect to see changes in the oscillator startup time and sensitivity.