What is the difference between congruency and equality? What is the difference between equality and congruency? When should I say that two figures are congruent and when that they are equal?

In geometry, a "figure" is a set of points in the plane. So, two figures are _equal_ if they have the same points. In other words, two equal figures are exactly equal: the same figure.

Congruent figures have the same shape and size (informally) but possibly different points.

No diagram is needed for this explanation.

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Part of the confusion between "equal" and "congruent" probably comes from the fact that congruence is an equivalence relation. In other words, congruence satisfies many of the properties of equality. So if you think of congruent figures as equal, your reasoning may well not lead you astray. However, equivalence is not quite the same as equality, so sloppiness here may have consequences.