You'll feel the tension of this disconnect. Things might be smooth enough between you at first, but as time goes on, you'll recognize how differently you assert yourselves. Your strong egos will cause misunderstandings between you; in essence, you'll be so concerned with your own needs and drives that you won't be able to listen to each other very well, so the needs and drives of your friend will fly right over your heads.
Tension and discomfort will increase whenever you're together. You'll start to notice a pattern of misfired energies. You might stop in the middle of a heated discussion and realize that neither of you knows what you're fighting about, much less why it matters so much! The way to get over this will be to slow down and start listening to each other. To some extent, you'll have to abandon your own egos in order to do this, which won't be easy; but it's the only way you'll be able to treat each other's needs as being equal to your own.