While tzitzit speak to our role in life and our relationship to everyday things, tephillin have contact with even higher elements, like G-d’s light and how we use it to help rectify the universe.
For G-d wants us to be awash with the light of His holiness from head to toe, to irradiate that out to the rest of the world, and to thus allow for a great deal of rectification. And we do just that when we don tephillin. (And not only do we do that then, but we’re taught that each element of our being reflects the light of holiness back to its Source then too, which allows for a complete circuit of give-and-take and a whole other order of rectification.)
The experience is known as “being called by G-d’s name” (see Deuteronomy 28:10) presumably because of the fact that so much of G-d’s light envelops us when we don tephillin and we’re in such close contact with Him then that our beings could almost be confused for His presence (though there’s certainly no real comparison). In any event, this is how it happens.
We’re taught that our soul most especially exhibits itself in our heart and brain, and that we’re thus to place tephillin in close proximity to both. So one tephilla (singular for tephillin) lays upon above our forehead while the other one is set upon our arm (which is connected to the artery that leads to and from the heart).
The head-tephilla allows the holiness we spoke of before to infuse the brain, and the arm-tephilla allows it to infuse the heart. Thus both our brain and heart — and the whole of our being, accordingly — are rectified in the process, awash in G-d’s holiness, and thus sanctified. There are many, many halachic details involved in the make-up, writing, setting, and maintaining of tephillin, but suffice it to say that each of them plays a unique role in the transmission of that holiness.
Now that should seemingly be done every day, seeing how much darkness there is in the world, but that’s not so. For we’re actually *not permitted* to don tephillin on Shabbat and Yom Tov. Since a higher, bolder light emanates from Heaven on Shabbat and Yom Tov than tephillin can provide; thus we’re already awash with it each and every Shabbat and Yom Tov, know it or not (which makes the donning of tephillin then superfluous, and while a superabundance of light and holiness might seem to be a good thing, it’s actually not).