In theory, they should not be different, but in practice they
are. Aristocracy literally means 'rule of the best'; bureaucracy is
a system where rule is done by appointed officials who generally
are well-trained in their field (good training is also called
'merit'). In theory, aristocracy should be meritocracy (which means
rule by those who are best qualified) as bureaucracy usually is,
but in history, aristocracy is associated with rule of the nobility
and privileged who earn their position not by ability but by
birthright. In some countries too (i.e.) former communist
countries) bureaucrats also exploited their positions to attain
things they did not earn. But in general, yes, the two are
different but they technically should not be.