In the Bible, ‘belief’ and ‘faith’ are the same thing (see the answer to ‘What is the difference between faith and belief?‘), so we can’t really say that someone believes but doesn’t have faith because belief is faith.
Having said that, the Bible does make a distinction between a living faith and a dead faith: a living faith is one that expresses itself in actions; a dead faith is one that doesn’t change the way you live. See James 2:14-26.
The Bible also says that people who have faith that God exists can also be ‘unbelieving’ if they don’t trust him but, instead, rebel against him. See Heb. 3:7-19, where the Israelites, who had faith that God existed, didn’t trust that he could look after them and, instead, rebelled against him.
Our belief, then, needs to be one that trusts God and results in action.