The main difference is that the plane triangle is on a flat surface while the spherical triangle is on the surface of a sphere. One consequence is that the angles of a plane triangle sum to 2*pi radians (180 degrees) while those on a sphere sum to more than 2*pi radians.
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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.
A plane triangle looks like a common triangle. A plane triangle is solved with linear units. A spherical triangle is found inside of a sphere. This type of triangle is solved with angular units.
Because, to allow for the curvature of the spherical surface, each angle must be slightly larger than its plane-surface equivalent.
ANY AND EVERY plane triangle totals 180o. Spherical triangles are far too complicated for this site.
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.