Question: What I mean is that religious Jews often use words like shabbos, mitzvot, teshuvah, siddur. Why not use the English words, Sabbath, laws, repentance/penance, and liturgal order (for example)?

After all, no two languages converge and represent each other perfectly or exactly, yet we usually submit to the compromise of [/the compromise that is] translation.

I feel like I'm missing something?

Thanks in advance.

Answer: 
Most of the users of this site are currently observing Pesach, so please do not be offended if you do not get an authoritative response for at least a couple of days. I am a Gentile believer, and am as such not permitted to observe Pesach, so I will do my best to ‘mind the shop’ whilst the more informed members of the community are away.

Firstly, you will see in Genesis (Beresheit) 1 that G-d uses the Hebrew language to name things - Day, Night, the Sky etc. - before ever a man was created. Note that in this chapter G-d is speaking ‘internally’: He is not addressing anyone outside Himself because these words were internal to G-d. It therefore follows that the Hebrew language is the language of G-d Himself, ‘in the beginning’: it has a special and preeminent status among languages. And if G-d says something is called X, then that is what it is, essentially.

You will see in Isaiah (Yeshayahu) 45 that Hashem, our god, speaks the Truth, and that He declares what is right. He did not say that English is the right way to speak of the essence of things. He declared, in using Hebrew to name the attributes of Creation, that the Hebrew language speaks of the essence of things. You will be aware of that famous question, ‘What is Truth?’. Jews will tell you that Truth is G-d’s seal. It is His thoughts, His wisdom, His understanding, that is Truth. And if G-d is unchanging (see Malachi 3), then Truth must be unchanging. Knowing the proper names of things is part of the Truth.

So, to answer your question, we use Hebrew words in the midst of English because we know that these are the correct ways of speaking of what we are speaking about. If one knows the correct way of speaking the Truth, but chooses to use another way of speaking, he is basically lying or deceiving. Using Hebrew words in the midst of English or any other language is a reminder that English is not a ‘final vocabulary’ and that we ought to be aware of the difference between Hebrew and other languages, and Hebrew speakers and other peoples.

You can find a much deeper explanation of the Jewish relation to Hebrew, on the one hand, and secular languages, on the other, in Part Three, Book One, of Franz Rosenzweig’s Star of Redemption (first published 1921 CE). This book may perhaps help you on your quest. Please let me know if you require any clarifications. Keep seeking.
Maybe similar to the reason Christians use the Hebrew words amen, hallelujah, hosanna. It is quicker than explaining them in English. The English terms aren't exactly equivalent.