Application to everyday life
Any time you watch the "spinning beach ball" or "hour glass" on your computer, trying to decide whether it's time to reboot or just wait a little longer, you are doing something like trying to decide the Halting Problem which has Turing degree $\mathbf 0'$.
On the other hand, calculating the bill and the tip at a restaurant has Turing degree $\mathbf 0$.
Application to physics
Random sequences (almost all sequences) have Turing degree incomparable with $\mathbf 0'$.
Imagine that you conduct a "Schrödinger's Cat" type experiment repeatedly, in fact infinitely many times. Supposedly the resulting sequence of dead/alive or 0/1 bits will be random. But how random? Turing degree theory allows for a classification: will the sequence be Martin-Löf random relative to $\mathbf 0'$? Relative to $\mathbf 0''$? Relative to all reals that are first order definable in the language of set theory? What does it really mean to say that at the fundamental level, the physical world is random (i.e., follows a probability distribution)?
On the other hand, apart from collapse-of-the-wave-function phenomena, it seems that the physical world is deterministic and even computable, i.e. Turing degree $\mathbf 0$. Is this really the case, or is the appearance just a consequence of our limited abilities to understand anything noncomputable?
Lack of applications
All this being said, experience has shown that computational complexity ($P$, $NP$, etc.) is more practically relevant than Turing degrees in many ways. And already in 1975, Baker, Gill, and Solovay showed that the methods used in Turing degree theory and computability theory more generally, being relativizable, can never settle whether $P=NP$. It kind of seems like 1975 was a watershed moment -- before that, Turing degree papers where more routinely published in Annals of Mathematics.