A paradox is a seeming contradiction, i.e. when one believes two statements which may superficially appear to conflict but, in the last analysis, can be shown to be compatible. A contradiction occurs when one believes two mutually exclusive statements are both true.
One might think it obvious that there must be a process one must follow in order to differentiate between paradoxes and contradictions, but in practice, laymen - and, unfortunately, "experts" too - will sometimes attempt to excuse themselves from explaining how their beliefs which at least are paradoxical are not actually contradictory. One common theological example is the tension between the doctrines of divine sovereignty and human freedom. Both are common beliefs which at least seem to conflict. Yet sometimes a pastor will, after alluding to this tension, simply encourage his flock to have faith that these are both true and trust that God has it all figured out.
The most obvious problem with this lax approach is that it gives heretics room enough to justify belief in anything. Paul might as well have said, "if Christianity is unreasonable, then our preaching is in vain and your faith is in vain." A faith which cannot be defended is a faith not worth defending. This isn't to suggest that a Christian ought to know everything. What it does suggest is that to justifiably believe any two doctrines which appear to conflict, the Christian must at least be able to present a speculative resolution compatible with Scripture.