'to deal ' with an issue implies that this particular issue is associated with difficulties ,that in order to have dealt with it , you have to overcome those difficulties . 'to address' an issue is the initiative phase of dealing with a problem/issue. It is to consider and think about this issue and begin dealing with it . That is why it is more often associated with business matters and sounds therefore formal. Other ways to say it is : to tackle a problem or an issue to get to work on some issue ..

These expressions can be used in daily life, but you're right that they're a bit businessy. 'Issue' in particular is business speak, and so is 'address'. 'Deal with' isn't especially businessy, so in everyday situations you might say 'deal with problems' or 'sort out problems'.