feature-class-to-feature-class... does the following
Converts a shapefile, coverage feature class, or geodatabase feature class to a shapefile or geodatabase feature class.
However, there are a variety of rules that apply to field mapping and joins that are applicable here that aren't elsewhere.
feature-class-to-geodatabase
appears to do the same, but the outputname is based on the input name
right-click only allows for the output geodatabase specification and the filename change
These are just base differences between the three. A read of the above links should be considered if you have specific requirements regarding the nature of the transfer, the destination and final properties.