This is not really a simple question to answer. There's no way to really define how similar things are over time. Additionally, the 1st-2nd centuries were a time of incredible amounts of change in the Jewish community, and neither ancient or modern Judaism are homogeneous.

The foundation texts that define modern Jewish practice were not written yet in the 1st-2nd centuries. The first text that goes into the canon of Rabbinic literature, the Mishnah, was compiled in the mid-2nd century.

The Mishnah does not seem to be a legal code exactly, but it is a compendium of opinions of things in Jewish law of sorts. Its content serves to illustrate some of the similarities and differences. For the similarities, the holiday schedule, basic form of the liturgy, practices like blessings on food, some elements of theology, etc are present in the 1st century. Though it's important to note that this is the similarities specific to the 2nd century forerunner of modern Judaism--these similarities would not be present for groups that did not evolve into modern Judaism. And of course some modern Jewish groups have altered these things, though they still exist in some form today.

But the content also shows how different the Jewish world was. Agricultural laws were an important part of the laws it contained, but not many modern Jews work in agriculture. States of ritual purity/impurity were a huge part of what the Jewish experience was in the Second Temple era and immediately after, but it is not today in the same way. The Temple and the laws and rituals surrounding it were important parts of Judaism then, but are only theoretical now, and only their study is part of the Jewish experience (and even this is limited compared to practical things) Jewish courts enforcing tort law is a major point of discussion, and while Jews do still sometimes have their financial disputes settled by Jewish law, this is not nearly such a significant part of what the Jewish communal authorities are doing.

Really, modern Judaism itself is similar to its ancient ancestor in important ways. But what the Jewish religious experience looks like is really drastically different, because life in general is. And that's only comparing to the predecessor of modern Judaism, we have much less textual material left over from other groups. And nothing in history is truly static, so Jewish law has developed over the past two thousand years. Depending on your perspective, that could be "very similar" or "very dissimilar".