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.
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.
In everyday language (outside of mathematics/geometry) we almost never use "equal" to mean that something is equal to itself. We use it to talk about two or more separate things, never just one. I was taught to say two line segments or two angles are equal if they have the same measure. In the "olden days" when I was coming along, only closed figures could be said to be congruent. Apparently many textbooks still read this way, for I tutor online and see many cases where students today uses "equal" and "congruent" in the "old" sense. I'd say anybody who gets confused between the two systems isn't really using their noggin.
two figures are congruent if their corresponding parts are of the same measurement. Ex. If two segments have equal length, then they are congruent. It is informal to say that two figures are equal. Two figures are not equal, they are congruent if the coreesponding measurements are equal.
Thomas Jefferson was wrong. All men are not created equal. They are created congruent. They may have equal rights, but no man is equal to another man.