Are you saying you pressed a Ride Stop when you saw a drive tyre / wheel about to detach...? Fair play for thinking ahead / convenience etc but I would really expect ride ops to be trained to hit E-stop for that kind of thing. You shouldn't need to put much thought into it... you just want the train to stop asap!
To answer the question about the differences, electrically, Ride Stop does, generally, stop motion - yes - but that is NOT the same as cutting power. There could be drives etc (VFD's) with power flowing through and the whole ride would still be under the control of whatever the software (program) is telling it. This is not considered a truly safe state. You can just restart while keeping in 'Auto' mode, which is less safe, operationally too. Better is for techs to switch to 'Maintenance' mode, confirm all ok, then restart.
Upon E-stop, the power is truly removed from all ride equipment. Motors etc go to something called STO (Safe Torque Off) and there's no software signal that can make them move. You can also perform maintenance on the ride in E-stop (you wouldn't dream of it in Ride Stop). In fact, rides are often locked out for maintenance by just pressing the E-Stop, flipping a cover over the button and locking so it can't be released. Having said that, there are in fact different levels of E-stop too. If you wanna go full nerd, you can read about that too...
https://www.se.com/eg/en/faqs/FA225420/
Hope that helps!