What is the difference between a line graph and a linear equation?

A linear equation, when plotted, must be a straight line. Such a restriction does not apply to a line graph.

y = ax2 + bx +c, where a is non-zero gives a line graph in the shape of a parabola. It is a quadratic graph, not linear. Similarly, there are line graphs for other polynomials, power or exponential functions, logarithmic or trigonometric functions, or any combination of them.