To put it much less formally than curiousdannii's excellent long and informative response.
The defining difference is indeed the Incarnation, but this is not only "an event" but marks a foundational difference in both the content and how it is perceived.
Between the two - many centuries of perceived "silence" - at least by the Jewish people of the day.
There's a little bit of history to the titles Old Testament and New Testament that needs to be explained.
We begin with the Greek word διαθήκη (diatheke) which has two major distinct sub-senses and cannot be translated with a single English word.
a last will and testament
a compact/contract/covenant, and in this sense it was used to translate בְּרִית (...
Pope Pius XII explains in his 1943 encyclical on biblical studies, Divino afflante Spiritu, §21:
this special authority or as they say, authenticity of the Vulgate was not affirmed by the Council [of Trent] particularly for critical reasons, but rather because of its legitimate use in the Churches throughout so many centuries; by which use indeed the same ...
The rationale is that since Jesus preached while the Torah was in effect, His teachings were part of the Torah and were fulfilled by His sacrifice on the cross and are not binding on us.
The motivation is to get around something that is condemned by Jesus in His teachings but which is not condemned anywhere else.