AMS vs LD vs ZLDO vs NCP in 1117 Linear Regulators
NCP1117


  A 10uF ceramic or tantalum capacitor should be adequate for
  most applications.


AMS1117


  The addition of 22µF solid tantalum on the output will ensure
  stability for all operating conditions.


The requirements are essentially the same, only Advanced Monolithic Systems are recommending a larger value to ensure stability under all conditions, whereas On Semiconductor provide a formula to calculate the minimum capacitance required. 

Unless for some reason you need to use the minimum possible capacitance, go with AMS's recommendation and then it should work with both brands (and probably others as well).

As to which is 'best' for your project, that depends on what your criteria are. Is one cheaper, more readily available, come in the package style you prefer? If you can't decide then either one should be fine.