Matthew indicates that Herod wanted to kill John, but did not as he feared the common people's reaction. Mark, on the other hand, states that it was Herod's wife Herodias who wanted John dead. (For the avoidance of doubt, this is a different Herod from the one who massacred the infants: he died earlier.)
This is Mark 6:19-20:
By contrast, this is Matthew 14:5:
So Mark has Herod rather enjoy John's company, and indicates that the malice came from Herod's wife. In Matthew, it is Herod himself who wants to kill John. As contradictions go it's hardly severe, and I leave dreaming up infallibilist replies as an exercise for the reader.
However, to academic Bible scholars, it's a location where Matthew, who used Mark's gospel as a source, has altered the story slightly. The real killer argument for this comes towards the end of the story... Herod allows his step-daughter one wish, and Herodias gets her to wish for the head of John on a platter. Here are Mark 6:26 and Matthew 14:9 respectively:
Both essentially the same, though Matthew has removed "exceedingly". This is fascinating, because Matthew has kept a sentence from Mark that is incompatible with a change he made. In Mark it makes sense that Herod would be grieved about killing John whose holiness he feared and whose company he enjoyed, whereas in Matthew it really doesn't.
Since the two verses in Matthew are incompatible - 14:5 and 14:9 - this indicates that he was partially copying from Mark, but was also making changes, and didn't notice that one of his amendments contradicted something he copied faithfully from Mark.
There are also many other arguments besides these passages for Markan Priority, the view that Mark was a source for Matthew and Luke. I should also note that this is not some fringe theory for skeptics. It's the mainstream view of Biblical scholarship, and the Oxford Companion to the Bible entry on the "synoptic problem" provided much of the substance of this page.