The biggest difference is in mechanical effects: in hydraulics, physical force is primarily a function of the system pressure and velocity of the flow rate. For electromagnetics, those are reversed: voltage controls the speed of a motor and current controls the force, for example.

They also deviate from the linear regime in different ways, which means the more interesting components have to be built completely differently to perform the same job. A one-way check valve and a diode are both governed by similar equations on a macro scale, but you’ll never be able to understand the internal structure of the valve by an electronic analogy or the design of the diode by a hydraulic one.

Part of that is getting comfortable with the maths, and the way that’s visualised is just equations, graphs and sometimes things like diagrams of fields or heatmaps. I work in RF so we do antenna simulations and things like that, and the software generates radiation pattern diagrams in 2D, or in 3D heatmaps showing the energy density. It just doesn’t work to, say, try and think of what would happen if an antenna was spraying water out or something.