How come Harry didn't recognize the Half-Blood Prince's handwriting?

Snape's handwriting changed a lot over the course of 20 years.

A relevant passage from the book HP and the Half-Blood Prince¹:

Although Harry had offered to share his book with both of them, Ron had more difficulty deciphering the handwriting than Harry did, and could not keep asking Harry to read aloud or it might look suspicious.

This likely wouldn't be the case if the handwriting in the book was identical or similar to that of the adult Snape, who'd been teaching both Harry and Ron classes for the last 5 years. If it was, they'd probably both be able to read it as easily as each other.

A person's handwriting can certainly change a lot from their teens to adulthood; you can see a real-life example in this answer from another SE.

In fact, it can change a lot from one situation to another even at the same stage in a person's life. I once had a teacher whose handwriting was almost illegible (roughly a horizontal line with the occasional wiggle) when she was writing on paper but perfectly clear and readable when she wrote on a blackboard.² I suspect she was making a conscious effort to write more clearly when she was at a blackboard in front of a larger audience. So either of the following could be the explanation:

• Snape's adult handwriting on paper looked similar to how it did when he was a teenager, but his writing on a board (which Harry and Ron were more used to) was very different. IIRC he never wrote very much on their homework when marking it, so it's plausible they never saw much of his writing on paper.
• Snape's natural handwriting was still as cramped and hard to read as when he was a teenager, but he made an effort to make it more legible for the sake of his students.³

² No pictures, sorry.

³ The Slytherin ones, at least.

Handwriting changes significantly, year to year. Worse for longer gaps. And not just age, but mental state.

Samples of [CBS "The Early Show"] co-anchor Erica Hill's handwriting over the years.

Medical studies (targeting both old age, and neurological issues) have confirmed this:

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.

Emotional state dictates how you write as well:

As it relates to Snape:

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).

One of the things in the Harry Potter movies is that everything has a subliminal message. Much like Snape's (J.K.R.'s) use of Flower Language to tell Potter he's bitter/sorry about Lily's Death. And he continues to feel this way his entire adult life. After all this time? Always. At the time of writing the notes in the book, he doesn't have this motivation yet.

_{Also, Harry is dense.}