Classification or regression? Which model is more accurate if I only care about being above or under the threshold?
Classification is the more direct approach and it will likely give better results. This is because the model's goal is exactly the same as your goal - i.e. predicting whether the price is above or below the threshold - and it will maximize this accuracy.

A regression model that is trying to minimize MSE, for example, could give you a model that doesn't accurately put predictions on the correct side of the threshold - which it doesn't know or care about. The model does not have the same goal as you do.

Let's construct a very simple example to see why this might be the case:

Training     Test 
x      y     x      y
1.0    20    1.8    ?
2.0    20    2.4    ?
3.0    200   2.8    ?
4.0    200   3.2    ?


Here I used the rule if x&lt;2.5: y=20, else y=200 to generate the data.

A maximum-margin classifier would get all of the test examples right, but a linear regression model fit on the training data would get all of the test examples wrong.