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 him, a few more contrasts are made obvious. The ghost tells Hamlet when it refers to Claudius:
Ay, that incestuous, that adulterate beast,
With witchcraft of his wit, with traitorous gifts,--
O wicked wit and gifts, that have the power
So to seduce!--won to his shameful lust
The will of my most seeming-virtuous queen:
O Hamlet, what a falling-off was there!
From me, whose love was of that dignity
That it went hand in hand even with the vow
I made to her in marriage, and to decline
Upon a wretch whose natural gifts were poor
To those of mine!
So lust, though to a radiant angel link'd,
Will sate itself in a celestial bed,
And prey on garbage.
It is clear from the ghost's rant that Claudius, in contrast to king Hamlet, is immoral, a traitor, a seducer. He is wicked and lustful. He dishonoured the family by seducing Queen Gertrude. The ghost makes it clear that Claudius is not the equal of king Hamlet for he is less of a man than the previous king had been. The ghost calls Claudius garbage and Queen Gertrude's lust drove her to him.
Throughout the play there is just praise for the previous king, even Horatio mentions his honour and bravery in the battle against the Norwegian king. Claudius however, schemes and plots, all for his own benefit. He has murdered his brother, denied Hamlet the throne and then plots against him. He uses others to do his ghastly deeds. In this regard he uses Polonius to spy on Hamlet, Rosencrantz and Guildenstern and then Laertes to plot his death.
Clearly, Claudius is a vile, disgusting, character - the complete opposite of the much-loved, dutiful and honourable king Hamlet.