I believe I read recently something to the effect of: there are some academics who do not view Christianity as having been borne from Judaism, but that Judaism and Christianity both arose concurrently.

So I think the sort of thing you're thinking of are arguments like that of Boyarin's Border Lines, which essentially argues that Christian and Jewish identities formed in opposition to each other, i.e. that the theological divide forced both groups to concretely define who "us" meant. Note that that is a significantly different thesis from:

I'm assuming that this refers to Rabbinical Judaism and Christianity, and it seems to imply that Rabbinical Judaism is as different from the Judaism being practiced pre-diaspora as Christianity is. Is this true? To what extent did Judaism evolve as a direct effort to differentiate itself from Christianity, if at all?

This bit is fraught with difficulty in trying to answer. Answers to this question are frequently polemic--in my experience, this tends to be from Christians trying to argue that Christianity, not Judaism, represents an authentic continuation of ancient Judaism, usually with assertions of radical Rabbinic innovations in more or less everything.

Perhaps the best way to answer this is to state that Rabbinic Judaism represents an ancient Judaism. Rabbinic Judaism does have precedents in pre-Christian beliefs. But it doesn't represent the totality of ancient Jewish practice by any measure. It's just that it's the only group that survived and still identifies as Jewish (maybe excepting the Karaites, though their origin is a bit murky and is off-topic). Some of these theological elements that were absent in Rabbinic Judaism's precursors did survive in Christianity.

However, there's no identifiable pre-Christian Jewish sect that Christianity identifiably succeeds. While some of the theology comes from different ancient groupings, there's not one group that Christianity can claim succession from. At least not one we know about today.

Personally, I find the thesis that Judaism developed its self-identity through its separation from Christianity to be at least somewhat convincing. the much more radical suggestion that Rabbinic Judaism developed in reaction to Christianity doesn't fly very well with me. There's very little evidence that Judaism saw Christianity as a huge theological development at first. Judaism was already very diverse, so the motivation for regarding Christianity as a particularly novel threat is lacking. Further, far from having a cohesive view of Jesus and early Christianity, there are only a handful (a dozen at most) early references to Christians, which are not responses to Jesus or the earliest development of Christianity, but are a response to Christianity at a somewhat later time (meaning that the time when Christianity registered was not in the first century). In fact, the number of Jews who became Christians in the first century was probably quite small. Certainly not enough to motivate a theological response.

Besides the polemic motivation I mentioned above, a lot of that claim comes from the assumption that Jesus must've been huge for Judaism, since he's present in Christianity and Islam. But Jewish theology only takes an interest later on. There's a joke common in /r/judaism, which has precedents in (hostile) interreligious dialog of Medeival Europe, where when asked about Jesus, Jews feign a lack of knowledge about him. Pretty much all Jews today know who Jesus is, of course, just as they did in the Middle Ages. The point is that Jesus has very little significance in Jewish texts, little enough to make the thesis that Rabbinic Judaism itself developed because of Christianity to be extremely suspect.

So while equating Rabbinic Judaism with ancient Judaism in general doesn't give the diversity of ancient Judaism its due, claiming such novelty for Rabbinic Judaism is not well supported in Jewish texts. While specific elements (conception of self-identity, how heresy is understood) may well be in reaction to Christianity, the existence itself really isn't attributable to that. And with the massive upheaval in Jewish life during the period (Roman rule, the end of Jewish political autonomy, the gradual collapse of Judean and the rise of Babylonian Rabbinic scholarship), attributing things to Christianity is a likely candidate of the post hoc ergo propter hoc (after, therefore because of) fallacy.

How different is Rabbinical Judaism from post-exile/pre-diaspora Judaism?

Well, everything changes over time. And "how different" is tough to quantify. I'd say that while things to change, it's change within a continuity, not a radical break. Rabbinic Judaism in, say, the 4th century is still discussing the same things its precursors were in the 1st century.

And without centralized temple sacrifice, what makes modern Jews "Jewish"? What's the justification behind no longer performing those rituals?

Well, without the Temple sacrifice is impossible, at least with an understanding of Judaism that's from the era we're talking. And the parts that aren't don't really work without the sort of religious infrastructure the temple had. The presence of a diaspora community that had existed without sacrifice long-term meant there was precedent for not having a sacrificial system when the temple was destroyed. And a staple of discussion of why Rabbinic Judaism survived but other forms of Judaism did not is that the focus on study and religious law beyond sacrificial ritual meant that this sort of Judaism had a "portable temple", a way of meaningfully interacting with the religion, that could be accessed without a central sacrificial system. And Judaism retained (through today) abundant ritual and liturgical references to the temple and its sacrificial system, so sacrifice as a concept was never completely abandoned, even if practice of it was.

sources:

Kalmin, Richard. "Jesus in Sasanian Babylonia" Jewish Quarterly Review, Volume 99, Number 1 (2009). A review of Shafer, Peter. Jesus in the Talmud. Princeton and Oxford: Princeton University Press, 2007.

Sim, David C. "How many Jews became Christians in the first century? The failure of the Christian mission to the Jews." HTS: Theological Studies 61.1 & 2 (2005): p-417.

Guttmann, Alexander. Rabbinic Judaism in the Making: A Chapter in the History of the Halakhah from Ezra to Judah I. Wayne State University Press, 1970.

Dunn, James DG, ed. Jews and Christians: The Parting of the Ways, AD 70 to 135: The Second Durham-Tübingen Research Symposium on Earliest Christianity and Judaism (Durham, September 1989). Wm. B. Eerdmans Publishing, 1992.

The crux of the matter is that, according to Josephus, there were three branches of Judaism. There was a "Pharisean" middle class sect with it's rabbis (it's lawyers and teachers of oral law). There was a "Sadducean" upper class sect which denied afterlife and the authority of other books than the pentateuch and there also was an underclass "Essene" sect which had tenets very reminiscent of those propounded by the Jesus dude in the Christian bible.

The argument is Christianity developed out of the Essene sect just as modern Judaism developed out of the Pharisee sect. The Karaite Jews were previously identified as a development of the Sadducean sect but there was some dispute starting with middle ages about that was based on some superficial differences between the Karaite theology and the description of the Sadducees by Josephus.

Of course after the destruction of the temple there arose some new Pharisean customs, such as wearing black to bemourn it, and other customs went defunct. Still contemporary rabbis do learn things only applicable if a temple is around.