Perhaps somthing of the original difference between the words lingers in modern understanding. Both words are broadly defined in contemporary English as having to do with right and wrong conduct. Character and custom, however, provide two very different standards for defining what is right and what is wrong. Character would seem to be a personal attribute, while custom is defined by a society over time.
People have character.
Societies have custom. To violate either can be said to be wrong, within its appropriate frame of reference. It is possible to draw an objective distinction between good and bad, right and wrong, which does not depend on context for a measuring stick. This is a standard which can be objectively tested for, in theory. While an objective standard of right and wrong may be desirable, this is not to say that objective standards and subjective standards are complete strangers to each other, necessarily.