axis equal sets the aspect ratio so that the data units are the same in every direction. The aspect ratio of the x-, y-, and z-axis is adjusted automatically according to the range of data units in the x, y, and z directions.
axis square makes the current axes region square (or cubed when three-dimensional). This option adjusts the x-axis, y-axis, and z-axis so that they have equal lengths and adjusts the increments between data units accordingly.
The easiest way to see this is to experiment with them by doing the differentaxis styles with the same data plot.
Briefly,axis normal uses the default aspect ratio for the plot,axis square makes the aspect ratio square,axis equal uses the same length for the data units along each axis, andaxis tight sets the axis box tightly around the data.
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