LTspice 60 Hz notch filter simulation
Nothing, the AC response analysis done by LT spice is correct. They have built a high pass filter. I've also never seen a notch filter built in this manner. 

Granted I spent 5 minutes trying to figure out the tutorial, and I can't. They never show the whole circuit. I'm thinking they analyzed it wrong or wrote the tutorial wrong. 

I don't know that the filter you have built isn't a notch filter without doing a transfer function analysis to double check (don't have the time right now), but LT spice is essentially the same thing. 

Why don't you do this, one up the tutorial and build your own twin T notch filter, you have two active amps so why not use both:



Source: https://www.changpuak.ch/electronics/Active_Notch_Filter.php

R4 and R5 change the gain and the other components are for the poles of the filter (a notch filter needs at minimum two poles).