why does it matter if Beethoven was Classical or Romantic?

If you are not interested in the role a composer played in the history of music, then of course it doesn't matter. However, some people ARE interested in how music evolved over the years. It just so happens that Beethoven was one of the most pivotal composers ever to have lived, providing a crucially important bridge between the old Classical period and the new Romantic era. Without Beethoven's innovation and influence, music would almost certainly have evolved slightly differently.

ALL categories are pointless in the end (is this Baroque or Classical or Romantic?). And remember that all these terms were coined artificially in the early years of the 20th century - Bach no more thought of himself as 'Baroque' (nor would he have even recognised the word) than Mozart considered himself a 'Classical' composer. These are all just terms invented by musicologists to make their categorsations easier.

EDIT: Just to respond belatedly to your last question. I have found that, even to music lovers who are not at all academic, putting a piece of unusual classical (small 'c') music into its chronological or historical context can be quite revealing. If one is reminded that, at the time Beethoven wrote his Eroica Symphony, it was the longest symphony (by far) that had been written up to that time, Haydn still had five years to live (and one or two pieces to write) and Schubert was a young boy of seven years of age (and Berlioz a newly-born baby), one can see why Beethoven's works were so revolutionary when they appeared.

It doesn't matter at all, I'm sure he didn't see himself as having entered entirely foreign grounds in music when looking at his teacher Haydn and other contemporaries. New composers of each generation after 1750ish took their new way of writing a little further without breaking away in any sense until Moderism (with the exception of Liszt, Wagner, etc.). Beethoven still remains a transitionary figure but in the sense that he is the transition from an early to middle period of an era, just like early and middle Baroque, rather than bridging two entirely different times such as the Romantic and Modern eras. The same question could still be asked, "Was Beethoven part of the early or middle period?" but I feel that in this new context his place in History becomes less torn between two times and therefore the question isn't enough of an issue to warrant a now arbitrary placing of him into one or the other period.

It matters because he is a bridge composer between the two periods. Music evolved during Beethoven's lifetime, and he was a prime innovator of the new Romantic period.

it doesn't matter really..people (society) just want to put a label on him (and everyone). It also makes it easier to learn(when your younger etc) that there are different periods of music and composers fit into them simple as....just complicates things for littleuns trying to learn th rudiments of classical music thinking about Beethoven pushing the boundaries and changing everything

Beethoven linked Classical and Romantic music, so there is a big debate about what he is.

It's mainly just a matter of dates and periods. There are some of his works that would be considered technically Romantic. As some of Paganini's solo works are technically Baroque.