If your question is simply: did Mohammed and Jesus both exist historically? Then the answer is, for both of them, yes. At least, the consensus among scholars regarding Jesus is that there was indeed a person, a Galilean Jewish man, who was baptized by John the Baptist and crucified under Pontius Pilate. Some scholars, notably David Strauss, and even some Christians (such as the Dutch Radical School) disputed the historicity of Jesus based on the fact that there is little to no description of Jesus as a human personality distinct from his mythical deeds. They are, however, a small minority.

Regarding Mohammed, there is no (serious) dispute about his historicity as an actual person. There is an abundance of material from non-Muslim sources who describe events of Mohammed's campaign. The Hijra (the flight from Mecca to Medina, and the beginning of the Muslim calendar) is considered to have happened.

But this question is hardly interesting in itself. In fact, if you ask me it is quite logical that these persons existed because it is generally easier for people to attribute stories to a person clouded in mystery than to fabricate stories with nothing to attach them to. It just does not instill the same kind of imagination in people. What we want to know is how the stories surrounding them compare in historicity.

Both Mohammed's and Jesus' main accounts of their lives, the Hadiths and the Gospels respectively, are considered to be wholly unreliable as separate sources. But the reasons for this are quite different. The Gospels are unreliable because (1) they were written about half a century after the events described and although the number of distinct accounts might suggest that there should at least have been some 'common core', the contrary is the case. The accounts are based on each other (Mark being the first whereupon Matthew and Luke based the account) and theorized to be based on the same outside sources (so-called hypothetical Q-source). Most importantly (2), the accounts are unreliable because the people who wrote the accounts did so from the perspective of Jewish prophecy. There was a long messianic tradition and a number of predictions present in the beliefs held by contemporary Jews. The lineage relating Jesus to King David is a common example, the Jewish expected the Messiah to be a descendant of David. But also his birthplace of Bethlehem and many of his deeds, both his plausible and miraculous deeds, are more or less fit into this picture of a predicted Messiah. Interestingly, according to one scholar, Bermejo-Rubio, some events that were initially not expected, notably the crucifixion, resulting in cognitive dissonance among early followers of Jesus, and consequently 'spiritualized' to fit the idea of the Messiah, leading to the resurrection and divinity of Jesus. Lastly (3) one must remember that in new religions there is always a blend of what is familiar (to give the beliefs justification) and distinct (to warrant a new religion). As such, we can quite safely say that in Jesus time there were a number of prophets, John the Baptist possibly being one of them, all of whom were grounded in the preceding Jewish tradition. Jesus being the most remarkable, due to who knows his charisma or his crucifixion, resulted in the separation of Christianity and its contemporary religious traditions.
Why then, do we know that Jesus did indeed exist? Well, there are a number of Roman sources that do point to a person named Jesus and his prosecution under Pontius Pilate. Tacitus is always named as one, as is Josephus. But to be honest, it is not one particular conclusive source that makes Jesus a plausible person. That Jesus was one of many prophets accounts for the fact that there is little direct mention in non-Christian sources. It is mostly a considerable amount of cross-referenced small accounts that convinces most scholars.

Mohammed's account, the Hadiths, is problematic mostly because initially, the tradition was purely oral. There was a requirement that each of the stories was preceded by a recollection of who told whom (tracing the story back to Mohammed) but these were quickly fabricated. The first written accounts date two centuries after the events described. Having a new hadith came with some prestige and thus the stories became unreliable quickly. Regarding what is historical about the accounts and why I can say very little. The first reason is that I don't know as much about Islam as I do about Christianity and don't want to go into speculation too much. The second reason is that even the more sceptical Western historians that I know of (such as Ignac Goldziher) although disputing the Hadiths, never questioned the Mohammed as a person. None of the works actually go into the non-Muslim evidence for that as they do for Jesus.

I hope I have answered your question, but do not hesitate to ask for elaboration if anything remains unclear. There are some references below but most of what I described is from memory (bachelor+master in comparative religion). If you doubt anything in particular I will retrace the literature upon request.

William Shepard, Introducing Islam, p. 35. (Goldziher's objections) Ole Davidsen, “Is there a Monkey in this Class”, Semeia: An Experimental Journal for Biblical Criticism (on the predictions inherent in Jesus' story) Fernando Bermejo-Rubio, “The Process of Jesus’ Deification and Cognitive Dissonance Theory” (on Bermejo-Rubio's mentioned theory)

Edit: lies to lives; decade to century