The tstat probe does NOT go inside the tub. It needs to be outside the tub between the uth and the bottom of the tub.
Having it inside places your snake in danger as if its moved, urinated on, or flooded with water it will tell the tstat to ramp up the power and over heat the uth giving the possibility of your snake getting burned.
All thermometers will vary in readings. The package should tell you what range of error is possible for the device. I would get it set up correctly and then see what the two give you.
The thermometer would be best on the bottom of the tub under the substrate. The substrate should only be a 1/4 inch at most to allow the heat to pass to the snake.
What feels warm to us is usually hot to them so don't worry about the hotspot not feeling warm to you. That's normal.