Harpooners are very similar to Anchormen in appearance except that they carry Harpoons instead of anchors. This makes them a very different fight however and when they are paired with Anchormen, they can be quite formidable.
Their main method of attack is throwing their large harpoons at you from a distance. They have less health and don't have the defensive abilities of the Anchorman so they are a lot easier to kill. For this reason, it may be in your interest to take them out first which will make fighting any anchormen easier. Just watch out for their harpoons and they shouldn't pose a problem.