The name Beelzebub refers to one of the seven princes of the underworld, or hell. The name Beelzebub comes from or is derived from Ba'al Zebub, Ba'al Z'bub, or Ba'al Z'vuv, all of which literally translate to mean Lord of the Flies. This deity was Semitic in origin, and believed to have been worshipped by the Philistine inhabitants of the ancient city called Ekron. Within these early polytheistic Semitic contexts, there doesn't seem to have been any large distinction between Beelzebub Lord of the Flies, and another god called Ba'al. Perhaps they were two separate deities that folded into one over time, or perhaps they were the same deity described with two different names. This god was named by the ancient Hebrews when they were discussing their Philistine enemies, and from these Old Testament references the term Beelzebub would eventually be picked up by early Christians, who would later use it as another name for Lucifer or Satan.
Contained within the apocryphal literature known as the Testament of Solomon, Beelzebul is referred to as the prince of all demons. Note that the spelling is Beelzebul, and not Beelzebub. In these noncanonical scriptures, it states that Beelzebul was previously an angelic general and prince in heaven, who was also associated with the Star Hesperus, also known as Venus, or the Morning Star. In these passages there is no debate as to who the author is referring to, as clearly it is the Satan and Lucifer we all know from the Old and New Testament of the Bible. The author goes on further to state that this character Beelzebul spreads chaos and destruction upon the earth through evil tyrants, and that he will also trick men into worshipping demons and false gods. Further it's said that he is the cause of lust amongst priests and men of God, the cause of murder, rape, and carnage throughout the world, and is also the sole bringer of war.
There are some contemporary Christian denominations that consider Beelzebub to be the ruler of Hell, and one of Lucifer's chief generals. It is claimed that Beelzebub stands in command of legions of demons, which he will ultimately lead into conflict during the final battle detailed in the Book of Revelation. What's unique about these contemporary Christian beliefs, is that the character Beelzebub has completely separated from the idea of Lucifer or Satan, and they or no longer one in the same. This belief that Beelzebub is a subservient deity under the command of Satan, is also quite common among any New Age occult that has heavy Christian influence. You will find quite the opposite belief the closer you get towards more orthodox or conservative Christian denominations such as Catholicism, who still largely consider the deities known as Beelzebub and Satan to be the same figure, as most canonical scripture seems to suggest.
What is strange, and I guess you could say quite hilarious, is that non religious secular media such as comics, art, and movies, seem divided as well with regard to whether or not Beelzebub and Satan are the same figure or not. My guess is that the authors of such media are not aware of the theological debate on the issue, and they simply reach their conclusion one way or the other by randomly scratching the surface of any particular Christian denomination, and then simply mirror what they find. Perhaps they aren't even aware of the difference that exists within the art media realm itself. You have to admit it is a strange example of art imitating real life, which in this case real life is actually imitating ancient myth. At any rate, my guess is that no one will care whether or not you equate Beelzebub with Satan, or classify him as a separate deity. The debate is quite trivial in the greater religious context, and isn't likely to cause a conflict of faith.