the first difference is that christians do not believe that the jewish scriptures are corrupt. if you were to look at the jewish scriptures you would find that repeatedly the scriptures testify that the majority of the jews would not accept what god was about to do (though jews would of course say that these verses speak of past groups etc.). furthermore, the very jewish scriptures speak of a new covenant which god would make with the jews and it would be unlike the one he made with them when he brought them out of egypt. this is referring to the mosaic covenant:
31 “The days are coming,” declares the LORD,
“when I will make a new covenant
with the people of Israel
and with the people of Judah.
32 It will not be like the covenant
I made with their ancestors
when I took them by the hand
to lead them out of Egypt,
because they broke my covenant,
though I was a husband to them,”
declares the LORD.
33 “This is the covenant I will make with the people of Israel
after that time,” declares the LORD.
“I will put my law in their minds
and write it on their hearts.
I will be their God,
and they will be my people. --- Jeremiah 31:31-33 NIV
a different word for covenant is testament. here god is saying that he will make a new testament with the children of israel. then notice the words of christ:
And he took bread, gave thanks and broke it, and gave it to them, saying, “This is my body given for you; do this in remembrance of me.” 20In the same way, after the supper he took the cup, saying, “This cup is the new covenant in my blood, which is poured out for you. 21But the hand of him who is going to betray me is with mine on the table. 22The Son of Man will go as it has been decreed, but woe to that man who betrays him.” 23They began to question among themselves which of them it might be who would do this. --- Luke 22:19-22 NIV
the new covenant which god promised is christianity and in the same way that the old covenant was ushered in with the blood of a sacrificial lamb (Moses then took the blood, sprinkled it on the people and said, "This is the blood of the covenant that the LORD has made with you in accordance with all these words." --- Exodus 24:8 NIV), the new covenant was also ushered in by the blood of the lamb who takes away the sins of the world. christianity has a clear continuation from judaism, in fact for the first few decades it was categorized as a jewish sect. we find clear evidence within the bible pointing to a new covenant that god would make with his people as it regards christianity but there is no such line of continuation when it comes to islam. muslims themselves see this and this is why they have to claim that the words of god were corrupted (even though the bible is very clear that god's word endure forever) because they cannot establish a line of continuity if we take the bible at face value. in fact almost every religion that tries to incorporate the bible within their beliefs have to claim that the bible was somehow corrupted etc. and thus we should not really look towards it but to the new teachings of their particular prophet (whether this be muhammad, joseph smith etc. it is all the same). so while the christian can affirm the whole torah and prove their beliefs from there, the muslim and most other religions cannot and therefore have to claim that the books are corrupted in order to answer for why they cannot establish a line of continuity.
that said, according to jewish scripture god did indeed come down as a man and even ate with abraham, so it isn't that shocking. and as it regards to paul and the law, what he said was actually a very jewish thing to say. just as paul said that gentiles do not need to accept the law of moses to be saved, the jews also say the exact same thing! as a jew paul followed the law of moses but precisely because this law was only given to the jews, he said that gentiles did not need to follow it. ask any jew, they will tell you the exact same things. according to judaism the law of moses is only for the jews. so your point is moot. anyway, the main point here is that muslims, mormons, bahai's etc. cannot establish a line of succession (nowhere in the new testament do you find any mention of another covenant that god will make with the arabs etc.) and as such they are forced to say that the bible is corrupt and as such we should instead turn to the teachings of their particular leader.