Here is a chart on some of the differences between knowing and showing God exists. Obviously, there is much more to it than what is displayed here.
Knowing God exists: There is a difference between knowing our faith is true though personal experience and sometimes what is called intuitive knowledge (i.e., something that is directly apprehended). Disciples of Jesus are blessed to receive the assurance of the truthfulness of our faith through the work of the Holy Spirit (Rom 8: 16-17; 2 Cor. 2:2). However, people of other faiths claim to have personal revelations/experiences. Thus, people have contradictory religious experiences that seem quite real. For example, Mormons claim that the Holy Spirit confirms their faith as true by a “burning in the bosom”—this is something they consider to be a confirmatory personal experience.
Showing God exists: While religious experience and intuitive knowledge or sometimes what Alvin Plantinga calls “Properly Basic Belief” is important, all experience must be grounded by truth and knowledge. Knowledge can be the key thing as to what keeps us close to God over the long haul. Plus, Jesus says we should love him with all our being (i.e., mind, emotions and will). Sometimes people think that personal religious experience negates the need for having other good reasons for faith.