What is the difference between Euclidean and Cartesian spaces?

Point in Euclidean plane can be written in many ways: either using Cartesian coordinate system, or polar coordinate system. That is same point $p$ can be written in two ways... If we are saying Euclidean plane, It simply means that we are giving some axioms and using theorem based on that axioms. But if we are saying Cartesian plane, it means that with euclidean axiom we are giving some method of representing of points.

This means: Euclidean Plane means we have only some set of axiom

Cartesian plane means Euclidean plane+ One fixed method of representing points.

The Cartesian system is Euclidean space with coordinates. The Cartesian Coordinate System unified geometry and algebra into one system of analytic geometry.

If you know MATLAB, A weak way of explaining it is:

clf;
ezplot('sin(x)'); %Euclidean
grid ON; %Cartesian

Euclidean

Cartesian