Spirituality as far as Islam is concerned, is an inward journey to God, through purifying the soul of bad traits and adopting good traits. Thus reaching your true human potential by following the footsteps of the Prophet Muhammad (may God's peace and blessings be upon him)
This means ridding the heart of the haram/prohibited.
For example if see someone will an expensive car you do not desire him to loose that blessing nor do you desire it for yourself. This is all internal and not external.
A Muslim can be a good Muslim without being a Sufi but a Sufi cannot be good without being a Muslim.