Say I have some qubit $|q\rangle = \alpha |0\rangle + \beta |1\rangle$. If I apply some unitary $U$, I get $U |q\rangle$; great. I can also think of $U$ as a change-of-basis matrix that maps the $|0\rangle, |1\rangle$ basis to the $U|0\rangle, U|1\rangle$ basis. If I just decide to measure the original $|q\rangle$ in that new basis, then all of my very limited knowledge about all of this would lead me to believe I get $U|q\rangle$. So is there any actual difference between the two situations? This seems like a well-known or obvious thing but I really don't know.
Application of a unitary transformation $U$ on a state $|q\rangle$ really leads to a new state $U|q\rangle$. What you get after a measurement is a one particular outcome of $U|q\rangle$ state because the measurement leads to colapse of the state wave function.
If you repeat measurement many times you will get probability distribution of possible outcomes of the quantum state.
Another difference is a reversability. While transformation of the quantum state is always reversible because the unitary matrix can be always inverted, the measurement is irreversible because of the wave function colapse. To reconstruct the quantum state you need to do many measurements of same state in different basis and employ so-called quantum tomography.