The first difference between the two king is expressed by Hamlet himself in Act 1, scene 4:
... that these men, ...
Carrying, I say, the stamp of one defect,
... Shall in the general censure take corruption
From that particular fault:
Hamlet is indirectly referring to Claudius in this instance for whom it has become customary to host celebrations at night, drinking wine and playing music in celebration of his achievements, i.e. winning the hand of the queen and becoming king.
When Horatio asks him if this is custom, Hamlet mentions that there is more honour in ignoring this habit than in following it. By implication he is suggesting that Claudius has a defect and is corrupt by following this ritual, whereas his father, by ignoring it, was not.
When his father's ghost appears and speaks to...
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