I'm not an EM guy, but you might find it easier to understand the words in for something like heat flow.
"Flux" is the total amount of heat flowing through a surface (measured as heat energy / second, i.e. power in watts)
"Flux density" is the flux per unit area, or sometimes the amount of heat generated per unit volume (for example heating something in a microwave oven, or heat generated by nuclear reactions).
FWIW, temperature is a scalar field (which is simpler to visualize than the vector fields in EM) and the direction of the flux is therefore the gradient of the temperature field. If you visualize a temperature distribution on a plane by drawing a contour map, the flux direction is at right angles to the temperature contour lines, and the magnitude of the flux density is higher where the temperature contours are closer together.