It is important, I think, because it exists. I think it is rather remarkable that any of these points of concurrency exist at all, but the fact that there are four of them is remarkable. It is also interesting to look at when the points are collinear , concurrent etc.
There is a nice theorem about the orthocenter, but I doubt that it is important. it is just pretty. Consider the three vertices of the triangle and the orthocenter. Select two of those vertices and the orthocenter. Draw the triangle. the vertex you left out is the orthocenter of that triangle. Not only is it pretty , but it's proof is accessible.
I think completeness is enough justification for inclusion. Pretty is even better.
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