In my experience within the Jewish community, I have found that many tend to equate orthodoxy and religiosity. She's religious," a friend might say, "she's shomer shabbat [i.e., she observes shabbat]." The implication of such a statement is, of course, that those who do not observe shabbat in a certain way are, by definition, not religious.
But just as religiosity and Christianity are not the same, neither are religiosity and orthodoxy. The idea that external indicators of faith -- whether that is manifested as wearing a yarmulke or a cross -- are stand-ins for actual religiosity -- that is, the degree of one's faith -- is fraught with problems.
One Orthodox rabbi, Rabbi Dr. Nathan Lopes Cardozo, wrote in an Israeli paper about this very issue in 2009 (emphasis mine):
All my life I am trying to become religious, i.e. genuinely religious, but so far I have bitterly failed. Oh yes, I am observant, even "very observant." I try to live by every possible halacha. It's far from easy and boy, do I fail! But that is not my problem.
My problem is that I don't want to be observant. I want to be religious, and that is an entirely different story.
Sure, living in accordance with halacha is certainly a crucial component of being religious, but it is not what makes me religious. To be religious is to allow God entry into my thoughts, my deeds, what I see and what I feel. It is to have a constant, intense awareness of being in His presence, seeing His fingerprints everywhere, and living up to that awareness.
I believe in God just as much as many of those who are so often described as "more religious" than I am. Yet, while I often wear a yarmulke on Shabbat and attend services regularly, I am not readily identifiable as what most would describe as "observant" (or, too often, "religious"). I observe Shabbat, but am not shomer shabbat.
This doesn't mean that how I relate to my faith is without responsibility, or that my faith has no implications for how I live my life. The real-world impact of my belief on my daily life is just as real, just as significant to me, as to the Orthodox who wear tzitzit or grow peyos. My faith has long played a role in my ethics, and, consequently, my politics.
Indeed, as Rabbi Cardozo pointed out, religiosity is not about observance (i.e. following the strictures of outward indicators of religiosity), but rather about spirituality and awareness of the presence of God. For me, that awareness is made manifest in my commitment to social justice. For others, that awareness may be manifested in other ways.
A person can be as observant as they want in their search for true religiosity or a spiritual connection, but that does not mean that they will find what they are looking for. It is, in fact, possible, for the non-observant person, who finds God in hikes through the mountains and uses technology on Shabbat, to be more religious than the Orthodox Jew who observes every stricture of halacha. At the end of the day, religiosity is a personal experience, and while guidelines may be of some help, they can guarantee no outcome.