What is it that equality and equivalence have in common, and how are they distinct. Please give detailed answer
The meaning of 'equality' and 'equivalence' rather depends on the context.
In logic and mathematics:
Equality is a relation between objects. It is the relation that every object has to itself and does not have to anything that is not itself. So to say of two expressions that they are equal is to say they represent one and the same object or have one and the same value. Philosophers usually use the word 'identity' for this relation, though confusingly mathematicians often reserve the word 'identity' for the equality of functions or for equality within the scope of universally quantified variables.
Equivalence is a relation between propositions or sentences. Two propositions are materially equivalent if they have the same truth value. They are logically equivalent if they have the same truth value under all interpretations.
In everyday use:
Equality is often used loosely to mean that two things share important qualities in common. What qualities and how important they are will depend on the context. This is the sense in which one might say: All people are equal under the law.
Equivalence is often used to indicate that two things can be substituted for each other without any significant difference. For example, This qualification is equivalent to that one, meaning that either will suffice to meet the criteria for a job or a course, etc.