Plane assumes a flat surface, where the angles of a triangle always add up to exactly 180 degrees. Spherical assumes a curved surface (such as the surface of the Earth, where the angles of a triangle add up to more than 180.
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There are many differences but perhaps the main difference is to do with the parallel postulate - or equivalently - the sum of the interior angles of a triangle. The sum of the interior angles of a triangle that is drawn on a spherical surface can be much greater than 180 degrees. One consequence is that given any line and a point that is not on that line it is possible to draw more than one lines through the point which will not meet the line.
Trigonometry is the study of plane and spherical triangles. Plane trigonometry deals with 2 Dimensional triangles like the ones you would draw on a piece of paper. But, spherical trigonometry deals with circles and 3 Dimensional triangles. Plane trigonometry uses different numbers and equations than spherical trigonometry. There's plane trigonometry, where you work with triangles on a flat surface, then there's spherical trigonometry, where you work with triangles on a sphere.
plane trigonometry spherical trigonometry
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The two branches of trigonometry are plane trigonometry, which deals with figures lying wholly in a single plane, and spherical trigonometry, which deals with triangles that are sections of the surface of a sphere.