Please note that I do not mean to offend anyone, I believe both Jesus and Muhammed were real people.
The historicity of Muhammad is far easier to establish because of his numerous family members and companions, almost all of which are attested in later sources that supplement eachother. His son in law Ali, wife Aisha, and grandsons Hussain and Hasan for instance have all been attested in later historic sources. Contemporary inscriptions commemorating the death of Muhammad's followers such as Uthman have been found in Arabia, matching Islamic accounts. The mark Muhammad left on Arabia and the people in his environment is simply too great and coherent to be fabricated.
The so called hadith (Islamic sources on sayings and actions by Muhammad) are, although in variying degrees, relatively sound historical sources because they contain virtually no supernatural allusions and have been transmitted by known chains of scholars. They contain many details about his life and character, dealings with people and even about his appearance.
Having said that however, there are few direct historical non-Islamic sources about Muhammad, and the ones that exist have doubtful origins. There is a supposed letter to Christian monks called the Ashtiname in Saint Catherine's Monastery in Sinai, Egypt for instance, but its autheniticity is disputed. John of Damascus, a famous Christian theologian and Church Father has written about Muhammad only several decades after his death, mentioning meeting people who met Muhammad.
I would say that in general it is very likely that Muhammad did exist, although some parts of his life are obviously not verifiable, such as his ascent to heaven and mystical transportation to Jerusalem on the 6-legged horse Burak.
As for Jesus, historiography becomes much more difficult, since the main sources for his life (the books of the New Testament and apocryphal Christian writings) are so full of supernatural references, inconsistensies and unlikely occurences that they are less useful to establish historical facts about his life. What they do provide us with however is what people transmitted and thought about Jesus life at the time.
The current historiographical consensus is that Jesus was an actual person in early 1st century Palestine, and that he was most likely killed under the Roman Governor Pontius Pilate (his governorship of the region is supported by Roman commemorative inscriptions). Further details about his life are very hard to establish, since the historicity of his followers is also contested. Although Paul most likely existed, he never physically met Jesus, and although the Book of Acts mentions Peter and the other Apostles, details are left vague and Peters fate as having been martyred in Rome is uncertain as well. Most importantly, none of them left attested writings, except perhaps Matthew and John, son of Zebedeus; who supposedly wrote a Gospel each (for obvious reasons the Gospels are not sound historical sources).
Non-Christian contemporary sources on Jesus are limited, restricted to occasional references to Jesus and Christians by historians Flavius Josephus and Tacitus.
Compare Flavius' mention of Jesus in his Antiquities:
Festus was now dead, and Albinus was but upon the road; so he assembled the sanhedrin of judges, and brought before them the brother of Jesus, who was called Christ, whose name was James, and some others; and when he had formed an accusation against them as breakers of the law, he delivered them to be stoned:...
There are no other known contemporary non-Christian sources on Jesus except vague references in the Jewish Talmud and Mishna, which were compiled after around the year 200.
In my view, the fact that so quickly after his death, Jesus' life was mythologized (Jesus probably died about 30 CE, the first Gospel, that of Mark, may have been composed as early as 65 CE), shows that Jesus ministry was probably short and shrouded in mystery, but left a profound impact on his followers because of his charisma, message, and probably his eloquence. It also indicates that his following was most likely predominantly illiterate and uneducated, since no known sources that were written during his life exist. Nonetheless, his following spread very quickly over geographical distances, since Paul writes to Christian communities ranging from Rome to Greece, Asia Minor and Palestine. It seems to me unlikely that a religious movement with such momentum and based on following a certain person wasn't inspired by a real life historical charismatic leader figure.
Source: am student of religious studies :)
Edit: I want to add furthermore, that Islamic sources on Jesus are obviously also not very useful to verify historical facts about his life, because they are mostly restricted to the Quran and the hadith, and most likely used earlier Christian and Christian Gnostic sources as reference material. The story in the Quran of a young Jesus forging birds out of clay/mud and breathing life in to them is als found in the apocryphal (Gnostic) Gospel of Thomas for instance.
Edit 2: I see now that I may have slightly overstated the historical reliability of the hadith. Although I still stand by my view that they are more useful for determining details about Muhammad as a person, I realize there are several "but's" (like some people have rightly pointed out), such as the time they were written in and ideological/religious reasons for keeping, editing or discarding hadith.
(http://gnosis.org/naghamm/gosthom.html)
Edit 3: here are 2 titles about the development of early christianity which are excellent and also provide valuable insights in the person of Jesus as opposed to the Christian conception of Jesus:
McCullough is the most comprehensive introduction to Christianity I have read so far, I would recommend it to everyone. The first chapters especially deal with the historical context of Jesus' surroundings. Chadwick is perhaps a bit more of an old fashioned scholar, but his knowledge and use of primary sources is unmatched and he analyzes church history with English precision. Both great reads.
Is this simply because there are more contemporary non-Islam records of Muhammad versus non-Christians records of Jesus?
Just for the record, there are no contemporary accounts of either Jesus or Muhammad. All surviving accounts of both historical figures are dated after their respective deaths.
Now it is true, that accounts referring to Muhammed can be dated closer to his life than the sources concerning Jesus ...
(The famous Byzantine passage known as the Fragment on the Arab Conquests is dated to 636 CE, a mere 3 years following Muhammeds alleged death. Compare this to the earliest possible non-Christian reference to Jesus: Josephus; who (very briefly) mentions him in his Antiquities of the Jews. This book however, was written some 60 years after Jesus is said to have been executed. There are earlier Christian sources, the letters of Paul, which are dated 20 years after Jesus' death.)
... but still it doesn't prevent similar problems arising when determining Muhammeds historicity when compared to Jesus.
First of all, it should be noted that the existence of both Jesus and Muhammed is generally accepted as a fact by all scholars concerned with either subject. Historicity means to what extent their lives can be reconstructed using the historic method. In this respect, in both cases the source material is rather dismal from an historians point of view. All biographies of Muhammed were composed at least a century after his death, the same holds true of the gospel of Jesus, the earliest of which was written 30/40 years after his death.
In both cases, the biographies constitute religious writings, which may not be the most reliable material. In any case the sources are rather contradictory. It's been remarked that if you compare all the surviving gospels; there are only three things they all agree on: Jesus was born, baptized and executed. In the case of Muhammed, there is the well known problem with his biographies, in which the later ones claim to know more about him Muhammad than the earliest ones; which is highly suspicious. Relatively basic facts of his life, like how many children he had, are highly contradictory within the source material. The hadith (sayings of Muhammed) have also been heavily scrutinized, with close to none seeming attributable to him.
However, the main reason why the historical debate on Muhammeds historicity may seem to lag behind when compared to Jesus; has to do with cultural and religious factors:
Serious scholarly efforts to examine Jesus' historicity did not start until the culmination of the enlightenment era. Only then, was Western society ready to accept people critically analyzing texts which had previously been doctrinal. Since then, there has evolved a balance or even separation between what people believe and what can be actually proven.
In the Islamic world, this separation of religious and scientific thinking, has not yet crystallized in the same way. A defining factor in this, is Islams claim that the Quran is the literal and final word of God; as apposed to Christian denominations; who tend to regard the gospels as divinely inspired; but not the literal word of God. Doubting aspects of Islam (of which Muhammed is of course a part) tend to be regarded with suspicion and tend not to be socially accepted by society as a whole in Muslim dominated countries.
This makes authors there reluctant to publish material which truly revises previously held beliefs and it is therefore no surprise that, by and large, examinations into Muhammeds historicity have come from Western academia. There are exceptions, for example the author Ibn Warraq has written many books about the historicity of Muhammed and the reliability of Islamic scripture; but his name is a pseudonym. He feels the need to write anonymously, for fear of being attacked or murdered. A situation similar to this for a Western, Christian or former-Christian writing revisionist material about Jesus is close to unthinkable in this day and age in the West.
If the Islamic world however, continues on its present path towards (very slow) secularization and acceptation of atheist or agnostic beliefs; it is highly likely there will eventually be a renaissance of research into Muhammeds life.
(Sources:
- In the Shadow of the Sword, by Tom Holland
- Early Islam, by Karl-Heinz Ohlig
and;
Q: A Reconstruction and Commentary, by Harry T. Fleddermann)
I wrote about the wikipedia page on the historicity of Muhammad page and why it's pretty trash a couple years ago here:
Some other threads that I've written on the sources for Muhammad's life and early Islamic history are also here and may be of interest:
https://www.reddit.com/r/AskHistorians/comments/i1zyx1/how_historical_accurate_is_the_quran/g01wej9/
As I recommend in all of these threads, the best thing to read on this topic is Robert Hoyland's paper Writing the Biography of the Prophet Muhammad: Problems and Solutions
Related question: can we verify the existence of ANYONE in the New Testament? Is there any historical record of a definitely real person saying they met anyone from the Bible? What’s the closest you can get?