Firstly, albeit my answer is not like @Alex A, but I respect his answer (and as a rule of thumb, there can be diverse viewpoints in different individuals).
Secondly, based on my research/perception, celebrating the birth of the Prophet (peace be upon him and his family) cannot be considered as a Bid'ah or a sinful practice, and even it can be a good/profitable practice. But why?
Such practices are not bid'ah, because bid'ah (innovation) does not include such matters; also, it is not injunction in the Quran and traditions. In the holy Quran, where does it mention that it is haram to celebrate the birth of the Prophet, and to be pleased of that?! Actually, to the best of my knowledge, we cannot call it Bid'ah when/if we don't attribute such celebrating to a kind of Islamic order, and factually we do it with the intention of being pleased of the birth of Prophet Muhammad as the apostle of Allah who indeed helped us to find the right way of Allah, and likewise as a sort of honoring/respecting.
On the other hand, according to the holy Quran (Surah A'raaf), honoring/respecting the prophet is regarded as a character of Mo'menin (believes/faithfuls), so, such celebrating of the Prophet's birthday can be regarded as a kind of honoring his importance/greatness when he is not available now. Consequently, even logically, such celebrating would be a type of keeping alive his greatness/name which in practice can be equal to honoring the prophet who is not currently available to us (from an aspect).
Of note, I found numerous great detailed related answers, but unfortunately were in Persian (Farsi), otherwise I would link them for more helpful information.
Even though we consider that e.g. there is difference regarding the precise date of his birthday (and seemingly especially between 2 dates), despising that, there doesn't seem to be any remarkable/big problem if we celebrate both of them or at least the one which can be more probable (based on every one belief/assumption). And actually the significant issue is its positive and constructive effect on the society which show a kind of honoring to high-ranking position of the apostle of God (albeit it would be definitely it is held in its the most exact date).