In OTL, Cleveland easily took the Democratic nomination, and only narrowly defeated Harrison
in the general election.
In ATL, the Democrats would have probably chosen David B. Hill, after a bitter convention fight, and he would likely have been elected (despite the close popular vote, you'd have to somehow change 80 electoral votes to reelect Harrison; and Hill will almost certainly carry New York).
One outcome: Hill probably won't re-appoint Theodore Roosevelt as Civil Service Commissioner (TR having unwisely campaigned for Harrison). Perhaps TR, now out of a job, takes his friend Bill Cody's offer to tour with the Wild West Show in Europe.