Aside from spelling, speech, vocabulary, and written language, how has Charlie Gordon gotten smater?

 
A fine question.
I would say first that the changes in spelling, language, and vocabulary are not changes in themselves, but symptoms or markers of the larger changes in Charlie's intellectual capacity.
 
That said, I'm going to be referring mainly to the original shorter version of the story as I address the other changes (rather than the later novel).
 
On March 23, Charlie starts to speculate in a more nuanced fashion about the future, wondering if Algernon might be smart for good.
 
On March 25, he shows a greater awareness of time (what takes time, and what saves time).
 
On March 28, he shows greater autonomy and will, pushing back against the doctors' instructions. This is continued soon after, when he looks up words on his own.
 
On April 6, Charlie beats Algernon for the first time. This indicates increased abilities in spatial reasoning and problem solving.
 
On April 20, he shows himself more able to evaluate social relationships, and notices his coworkers' mockery. He follows this with observations showing greater self-awareness.
 
By May, Charlie is able to make comparative judgments of others' intelligences.
In other words, his intelligence improves in multiple areas at once.