The books of Exodus, Leviticus, Numbers and Deuteronomy contain a very strict and specific set of laws. Why don't modern Jews live as strictly to the scriptures and the ancient Israelites did?
I don't want my answer to come off as insulting, or personal, but my answer has to address the reasons behind your question, as many Christians ask us many versions of this same question, all of these questions stemming from the same thought process.. It's been my observation that Christians tend to feel like they understand the Bible better than the Jews do, or at least know more about it than the Jews, and feel they are in a position to point out when Jews aren't doing something that "we are supposed to be doing."
In your comments you said one of the things we "should" be doing is animal sacrifices. To you it seems clear that we as Jews are obviously in violation of the laws of animal sacrifices. The reality is that you are in ignorance, because the Bible clearly states that it was God Himself through His teacher Moses who specifically forbade us from doing animal sacrifices. Deuteronomy 12:5-14
5 But you are to seek the place the Lord your God will choose from among all your tribes to put his Name there for his dwelling. To that place you must go; 6 there bring your burnt offerings and sacrifices, your tithes and special gifts, what you have vowed to give and your freewill offerings, and the firstborn of your herds and flocks. 7 There, in the presence of the Lord your God, you and your families shall eat and shall rejoice in everything you have put your hand to, because the Lord your God has blessed you.
8 You are not to do as we do here today, everyone doing as they see fit, 9 since you have not yet reached the resting place and the inheritance the Lord your God is giving you. 10 But you will cross the Jordan and settle in the land the Lord your God is giving you as an inheritance, and he will give you rest from all your enemies around you so that you will live in safety. 11 Then to the place the Lord your God will choose as a dwelling for his Name—there you are to bring everything I command you: your burnt offerings and sacrifices, your tithes and special gifts, and all the choice possessions you have vowed to the Lord. 12 And there rejoice before the Lord your God—you, your sons and daughters, your male and female servants, and the Levites from your towns who have no allotment or inheritance of their own. 13 Be careful not to sacrifice your burnt offerings anywhere you please. 14 Offer them only at the place the Lord will choose in one of your tribes, and there observe everything I command you.
Your presumption that we don't do things has little to do with Jews not following the law of Moses. It has to do with your ignorance of the law of Moses, and your presumption that you truly know what the law of Moses says (because otherwise you wouldn't feel you were in a position to ask the question the way that you did).
Another of your other claims is that Jews don't follow clothing laws. Most observant Jews do live by the clothing laws mentioned in the Bible, therefore many refrain from mixed fabrics. But i can assure you that many other things that you think Jews are shirking are really just either your personal ignorance, or ignorance caused by translations.
One example is that many Christians ask why Jews don't wear blue threads in their tassels anymore. The reality is the normative Hebrew word for blue isn't used in the Bible regarding tassels, but rather a SPECIFIC word for a SPECIFIC type of blue. When the tradition of how to make that specific shade of blue got lost, Jews stopped wearing blue. Imagine if your English Bible said "and the children of Israel shall put a string of Prussian Blue on their tassels." More than likely, you would assume that this string needs to be that exact color of blue. But because your English Bible just translates the word as blue, it appears to you that any blue will do. Thankfully, now that some claim to have rediscovered the specific shade blue, it is starting to appear again and be worn by Jews.
By no means am I insinuating that you aren't intelligent. But the Bible is a vast book, and the history of its people is larger still. There are many chapters dedicated to animal sacrifices, but only one or two verses that show you can't sacrifice without a temple. So it's a mistake to do a cursory read of the Bible and assume that one has grasped it. Rather, one has to study these specific laws in very large detail before one can make assumptions about how these laws are being practiced or interpreted.
You probably still have many other "but what about X" questions in your mind regarding the practice of Jews, or possibly some assumption that we care about following Rabbis more than God. You should not approach the questions from that viewpoint, but should rather think, "I wonder why Jews don't do x the way I assume it should be done." Because that thought puts everyone on the same plane and allows for open dialogue. The majority of the time Jewish practice has to do with a nuance in the text that Rabbis picked up on, or even a direct verse from the Bible you hadn't read. I'm not saying we are perfect, nor am I saying that we always follow God above following Rabbis. But the majority of the time, you will find that we are following the laws of Moses, and we are doing so because we love God.
For a partial answer. The Orthodox Jewish Community does follow the Torah and its statutes.
Orthodox Judaism is the approach to religious Judaism which subscribes to a tradition of mass revelation and adheres to the interpretation and application of the laws and ethics of the Torah as legislated in the Talmudic texts by the Tannaim and Amoraim. Orthodox Judaism includes movements such as Modern Orthodox Judaism (אורתודוקסיה מודרנית) and Ultra-Orthodox or Haredi Judaism (יהדות חרדית)
Hope this is helpful.