The definitions for both these terms are different, so it seems to me that they are not same. But I wasn't able to find an example which illustrates the difference between these two.

Linear polarization diagram

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Linear polarization equation

Circularly polarized wave have their magnitude of electric field vector constant, but their phase change periodically as a function of time.(i.e. the electric field vector rotate periodically as a function of time)

Linearly polarized wave is same as plane polarized wave. Why we call it linearly polarized ? Because oscillation taking place along a line. Why we call it plane polarized ? Since oscillation taking place along one axis & the wave is forward in motion along another axis. Two axes are used , so called plane.

Image

Polarization

Circularlypolarizedlight

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Linearly polarized wave Oscillate in a single plane and the magnitude of electric field vector varies sinusoidally as a function of time.

So what exactly is the difference between a linearly polarised wave and a plane-polarised wave? Because according to me, a linearly polarised wave will oscillate in only one plane.

Since each point on the string moves on a straight line, the wave is also referred to as a linearly polarised wave. Further, the string always remains confined to the x-y plane and therefore it is also referred to as a plane polarised wave