The difference between Auxiliary bishop and Coadjutor bishop
When used as nouns, auxiliary bishop means a bishop who functions as another bishop's deputy, where the weight of sacramental duties is too much for a single bishop, whereas coadjutor bishop means an auxiliary bishop with the right of succession, i.e., he becomes bishop upon the death, removal or retirement of the incumbent bishop.
check bellow for the other definitions of Auxiliary bishop and Coadjutor bishop
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Auxiliary bishop as a noun:
A bishop who functions as another bishop's deputy, where the weight of sacramental duties is too much for a single bishop.
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Coadjutor bishop as a noun (Roman Catholicism):
An auxiliary bishop with the right of succession, i.e., he becomes bishop upon the death, removal or retirement of the incumbent bishop.