These two important characters seem to share many similarities. Both are diplomatic, charismatic speakers who easily stand out from the other fallen angels and show an ability to sway their audience through their powerful rhetoric, as Book II displays. Both are fallen angels, and Satan, in his opening speech in Book I, draws a parallel between their characters, almost describing them as equals:
If he Whom mutual league,
United thoughts and counsels, equal hope
And hazard in the Glorious Enterprize,
Joynd with me once, now misery hath joynd
In equal ruin...
We can therefore infer that both characters are fairly similar in terms of their character and their aims. Both united together in their attempt to try and overthrow the powers of heaven, and both share a similar fate as they are cast down to hell.
The key difference between these two characters however seems to be the way in which Beelzebub appears happy to be Satan's "second-in-command" as opposed to trying to seek power himself. He therefore appears to lack the same kind of driving ambition that causes Satan to continue his opposition to God even from his present location. He exhibits tremendous loyalty and faithfulness to Satan. Note how he addresses his chief:
O Prince, O Chief of many Throned Powers,
That led th' imbattelld Seraphim to Warr
Under thy conduct, and in dreadful deeds
Fearless, endanger'd Heav'ns perpetual King...
This, then, is perhaps the key to understanding the differences between them. We would never expect such servile terms of address to be uttered by one such as Satan. Beelzebub distinguishes himself from Satan through his willingness to be led and his loyalty.