Handwriting changes significantly, year to year. Worse for longer gaps. And not just age, but mental state.
Wills signed by elderly people are often contested on the grounds the the signature is different from their earlier specimen signatures. Neurological disease, which can affect handwriting, is very common and progressive amongst elderly people. Handwriting change due to old age and neurological disease is poorly understood. To better understand this subject, we carried out a large methodical study based on almost 200 handwriting specimens of Parkinson patients and age-matched controls. Interestingly, our findings indicate that some of the handwriting changes which occur in these populations tend to resemble forgery indicia although upon close inspection they are distinguishable from them. Thus, document examiners are urged to exercise caution in assessing purported forgeries on wills and other documents signed of written during older age or a writer suffering from neurological disease.
20 years is significant enough that Snape's ~40 year old adult writing, most often seen on the board and not on parchment, would be remarkably different from Snape's ~15 year old teen age scribbling.
As emotional outlook goes, teenage Snape was still a bit optimistic compared to post-Lily-ended-friendship, or post-Lily-got-married-and-had-a-kid, and especially post-Lily's-Death. Super morose adult Snape would not have slightly optimistic teen Snape's emotional state, so his handwriting would change (ignoring age related changes).
Also, Harry is dense.