Exalt, magnify, aggrandize are comparable when meaning to increase in importance or in prestige.
Exalt and magnify also come into comparison in their older sense of to extol or to glorify.
In modern general use exalt retains its implication of lifting up but emphasizes a raising in a scale of values without necessarily affecting the quality of the thing raised. Therefore one exalts something above another or at the expense of another.
Magnify stresses increase in size; it commonly suggests an agency (as an optical device) which affects the vision and causes enlargement of apparent size or one (as a vivid imagination) which affects the judgment and leads to exaggeration.
Aggrandize emphasizes increase in greatness or mghtiness; it implies efforts, usually selfish efforts, directed to the attainment of power, authority, or worldly eminence.