One way of seeing the physical difference between vectors and covectors is the way they scale when you rescale your coordinates. For example, if I change my distance units from meters to centimeters, then a $\Delta x$, expressed numerically in terms of its components in a chosen basis, goes up by a factor of 100, while a wavelength goes down by a factor of 100. This tells us that displacement is a vector, while frequency is a covector.
This is why gradients are covectors. For example, the gradient of a temperature tells you how much the temperature changes per meter.
The physical distinction between vectors and covectors is often obscured by the fact that if we have a metric, we can freely raise and lower indices. That could give the impression that the distinction is optional or unimportant. But there are practical situations where you don't have a metric, and then the distinction is mandatory and important. Here is an example. If you really want a thorough and careful discussion of this sort of thing, the book to look at is Spacetime, Geometry, Cosmology by Burke.