They are synonymous, really. There is one difference, however. A delay is more spontaneous in a way than a postponement. A delay can occur at random while something is usually planned to be postponed. People are given fair warning when something is postponed, while someone can be delayed in traffic without warning.

Well, both words have different meaning. Delay means that whatever event was expected to happend at a certain time will not take place at that time but later, for mainly negative reasons, usually beyond control. Whereas Postpone (from Latin meaning "puttin off for a later time") usually implies that the event that was programmed for a certain time has been put off in behalf of somebody's best interest. Basically there are negative connotations attached to daley, same as there are positive connotation attached to postpone.

delay:These verbs mean to cause to be later or slower than expected or desired. eg:The bus was delayed by a cloudburst. postpone:It's formal.The central meaning shared by these verbs is “to put off until a later time eg:postponing our trip