So since my high school algebra tells me that $361°$ is basically rotating the whole axis around and adding $1$ more degree, I would assume that at least in trigs they are the same.
But then $1°$ is acute, while $361°$ is not classified as either acute, obtuse or right. I understand this difference.
Then just out of curiosity, what other differences (any subtle differences would count) exist between $361°$ and $1°$?
In parametric equations they behave differently. For example in polar coordinates you can define a spiral by $r=\omega$, $\theta=\omega$ for $\omega\in [0,\infty]$.
$\arcsin(\sin(361^\circ))\neq 361^\circ$ whereas $\arcsin(\sin(1^\circ))=1^\circ$ and a similar problem will occur with the other trig functions.
If you only care about the final position (after a rotation) they are the same. If you care about how you got there (perhaps if you're winding string) then they are different. The qualities "acute", "obtuse" and "right" usually apply only to angles, not rotations. You don't usually ask about $361^\circ$ in that context.
The angles $1^\circ$ and $361^\circ$ describe the same geometric angle. That is, we say that these two angles are "coterminal". There are many times when it is safe to say that these are "the same angle", but it is occasionally useful to think of them as different.
For example, if we want to describe how an object rotates. The statements "the wheel rotated $361^\circ$" and "the wheel rotated $1^\circ$" are very different.
