As far as i can tell, both LQR and PID controllers can both be applied to the cart-pole (inverted pendulum) problem. What are the pros/cons to using one controller over the other for this particular problem? Are there any reasons/situations where I should prefer one over the other for this problem?
You are correct in claiming that both PID and LQR can be applied to the linearized form of the cart-pole problem. If you have a look at the answers to the related question Ugo linked to in the comments (specifically Ugo's own answer: https://robotics.stackexchange.com/a/5231/10414), you will see the high level differences between PID (and other classic methods) and LQR (and other optimal methods) for control.
For a more detailed analysis specifically focused on the cart-pole problem, an article was written by Nasir et al which specifically deals with this question, available as a cached PDF on CiteSeerX: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.545.6096.