Also wondered that. Looks like it was based on how the French transliterated the names of native groups and then how Americans ended up pronouncing each one.

Kansas comes for the native group called the Kansa. The French added a consonant on the end, which is not pronounced in French. Seems the French added that final "s" to indicate a people rather than individual, like les Illinois. So in French, Kansas meant "the Kansa people" and was pronounced "Kansa".

In English, that final "s" is pronounced, so we have Kansas.

Arkansas originated with the French transliteration of the Quapaw native group. The Quapaw and Kansa were originally part of a larger group that split when it migrated down the Ohio and Mississippi rivers, so neighboring tribes referred to them all as "akansa" or "arkansa".

So the French called the Quapaw branch "Arkansas" without pronouncing that final "s". This time, however, when English speakers predominated, they retained something closer to the French pronunciation, leaving us with "Ar-kan-saw".

Apparently it kind of went back and forth for a bit. In 1881, the Arkansas state legislature put an end to the debate, declaring "Ar-can-saw" the official pronunciation, my guess would be to differentiate them from Kansas.