Historically (and by historically I mean "in Euclid's Elements") the word "equal", when applied to geometric figures, meant "equal in magnitude". So for example:
- Euclid refers to two segments as equal if they have the same length
- Two triangles are equal if they have the same area
- Two solids are equal if they have the same volume
For example, Elements Book I, Proposition 35 says:
Parallelograms which are on the same base and between the same parallels are equal to one another.
(See figure below, from Fitzpatrick's English translation of the Elements, based on the Greek text of Heiberg.)
Notice that "equal", in this usage, does not mean "congruent"; nor does it mean "the same, identical". With this idea as background, the word "congruent" plays an important role: it allows us to say that two figures are not only equal in area but are precisely the same shape, i.e. can be superimposed onto one another.
Language, of course, is constantly evolving. At some point people stopped using the word "equal" to mean "equal in area", and began using it to mean "the same mathematical object". I am not sure exactly when this switch happened, but I suspect it was in the latter part of the 19th century, as set theoretical ideas began permeating all aspects of mathematics. The greatest sea change in Geometry during this time was Felix Klein's Erlangen Programme, which sought to reframe Euclidean Geometry as the study of properties that are invariant under isometries of the plane. In this context it is important to distinguish between the statements "$\triangle ABC = \triangle XYZ$" (which means that the two triangles are the same mathematical object) and "$\triangle ABC \cong \triangle XYZ$" (which means that there exists an isometry mapping one triangle onto the other).
(This is also when the partitional classification of quadrilaterals found in Euclid began to gave way to the hierarchical classification scheme most of us are more familiar with.)
The effect of this linguistic switch was that the relative valences of the words "equal" and "congruent" became reversed: whereas in Euclid's work the word "equal" is a weaker relation than the word "congruent" (in that congruent figures are always equal, but the converse is not true), nowadays the word "equal" describes a relationship stronger than congruence (in that "equal figures" are (trivially) congruent, but the converse is false.)
So there is one reason why it is important to have different words: to distinguish between different notions of "same" ("same area" vs. "same figure" vs. "same size and shape").
I would go even further, and say that a lot of the vocabulary we use in Geometry is introduced to disentangle notions that, to a naive student, seem equivalent.
- For example: to many students, a statement like "rectangle $ABCD$ is bigger than rectangle $PQRS$" seems perfectly sensible. That is because they have an unexamined notion of "relative size". But it is possible for one rectangle to have a larger diameter than another, and simultaneously a smaller area, and equal perimeter! (Which is larger: a $6 \times 8$ rectangle, or a $7 \times 7$ rectangle?) So we introduce vocabulary (area, perimeter, diameter) in order to unpack these distinct notions of "size".
- A second, more sophisticated example: if you ask students to find a point in the interior of a triangle that is equidistant from all three vertices, they will say "in the center". If you ask students to find a point in the interior of a triangle that is equidistant from all three sides, they will also say "in the center". It seems both obvious and unproblematic to them that "center of a triangle" means something -- and that it means one thing. But in fact there are multiple notions of "center", all of which are worth naming, and therefore we need different names (circumcenter, incenter, orthocenter, centroid) to distinguish them.