In theory none. They are the exact same thing. At least they claim to be the exact same thing.
1. Most christians learn Torah through their native language translation instead of hebrew.
2. English speaking christians only have 2 words to describe God. YHWH (I don’t know the vowel, I don’t want to know), Adonai is translated as Lord. El and Elohim is translated as God. Lord in english is not even a word for God. Landlords, for example, are typically not divine. In south east Asia the word God is translated as Allah and the word Lord becomes Tuhan, which is cognate with non divine lord, “Tuan”.
3. While aware of Jewish’ tradition of hiding God’s only personal name from ever being uttered out of respect, christians are not familiar with the “extra mile” approach Jews go through. Jehovah witness are christians’ sects that want to make YHWH name famous again.
4. Some actual differences in the torah due to actual different sources. This is more of a difference between what people think is the original rather than differences between Jewish and Christian tradition. Sample is Deuteronomy 32:8-9 where christians translation sometimes use the dead sea scroll and septuagint that use children of God rather than children of Israel. It’s a very important verse because it decide whether YHWH is the God of all universe or a mere God of a small nomadic tribe.
5. Actual vagueness in the Torah itself give a certain leeway in translation and interpretation. The word elohim, օրինակ, is a homonym that can be (correctly?) translated as God or angels or power. Okay, maybe I am wrong here. But that as far as most christians go when it comes to understanding torah given our language limitation. English translation tend to hide the controversy. Քրիստոնյաներ, are more “open” to the idea of anthropomorphic God. Ի դեպ, the idea that God incarnates at least once is the central theme of christianity. You can see discussion here: https://sites.google.com/site/yahwehelohiym/the-messengers/jacob-wrestles-with-a-man and http://www.mayfieldsalisbury.org/files/SermonSunday17thOctober.pdf. Most christians believe that God is among the 3 men that Abraham saw. Jews do not like that interpretation. Christians are somewhat confused. Atheists like that most because they want to show how absurd the whole thing is. Most christians do not know that elohym is homonym. Most christians do not even read the bible let alone dig into what’s behind the english translation. If the translation says Jacob wrestle with God, then God it is for most christians.
6. Theological differences. Christian do not think torah is that important. They think Jesus sort of cancel that out or something like that. Most christians will not say, Jesus cancel the torah though. So a lot of twisted logic shows up here. Jesus himself said that not one iota of Torah would go away till the earth and sky goes away.
7. Christians do not think Moses is that important. Deuteronomy 34:10 doesn’t apply. So “Since then, no prophet has risen in Israel like Moses, whom the LORD knew face to face,”. To christians this doesn’t apply to Jesus because Jesus is not just mere prophet. Most christians believe that Jesus is God and that faith is what often define christianity. Note that deists, arians, unitarianism are also branch of christianity that don’t believe Jesus is God. I am not familiar with messianic jews. Curiously the muslim also claimed that Deuteronomy 34:10 doesn’t apply because Muhammad is not Israelites.
8. Neither jews or christians practice politically incorrect aspect of Torah. They both have a different reasons. Christians think that an eye for an eye means literally that but somehow got canceled by new testaments. Jews seems to think that an eye for an eye and stoning people to death means something far more humane like monetary compensation (plus massage?).
9. Christians do not have oral torah. I am not sure whether christians will still believe written torah is inerrant if they saw the oral part. It’s as if in, if you think splitting the sea is crazy enough.
10. Many christians believe that torah is inerrant, somehow. That despite many differences between septuagint, masoretic, and dead sea scrolls and despite the fact that all the best bible expert in the word cannot figure out which version is the correct ones.
Difference between Torah in Christianity and Torah in Judaism is a post from: Free Market Forever