Cleveland has gone through its share of problems in the past several decades that even I - a Cleveland enthusiast - cannot deny. Columbus has been rather spared from some of the problems Cleveland has faced, and there is probably a little bit of envy. For example Cleveland has suffered some embarrassment and economic loss from its unnattractive manufacturing sector of the economy. Columbus never quite had the same industrial pressence and as a result they don't seem to have to shed this gritty reputation. The city of Cleveland has been slowly losing its population, while the city of Columbus has been slowly gaining its population.
The combination of those two things really concerns dedicated Clevelanders, because Columbus is increasingly viewed as compitition for new people, jobs, media coverage, conventions, etc. Cleveland isn't used to such nearby competition. Greater Cleveland has always felt uncomfortable being inferior to New York and Chicago, and now the possiblity of being inferior to Columbus in the future understandably really upsets or confuses a lot of people because of Cleveland's historical supremacy
Another reason why people get upset is that the statistics about Columbus being bigger and growing while Cleveland is smaller and shrinking are a bit misguided. Land area has a lot to do with it. Cleveland proper is very small in terms of land compared to Columbus proper. Cleveland proper is very small and when talking about 'Cleveland' one should really take into consideration the nearly fifty seperate municipalities within just twenty miles of Public Square (the heart of downtown Cleveland) that would put Cleveland well beyond the population of Columbus. Political geography in Greater Cleveland is very fragamented and always works to its disadvantage - especially in communicating the true population, median income, and new job creation.
Dedicated Clevelanders might also contest that Columbus has a greater white collar image than Cleveland. One might concede that a greater percentage of Greater Cleveland's workforce is blue collar, but the facts would still communicate that Cleveland has more white collar jobs than Columbus.
I admit that I don't know Columbus extremely well, however I don't really believe that too many amenites could be better in Columbus. Greater Cleveland is first of all a larger market that is able to support and sustain a lot of amenities that Columbus cannot. Second of all Cleveland boomed into a major city well before Columbus and has had a head start on a lot of things that take a hundred years to build up like parks, hospitals, museums, housing stock, diversity, etc.
So I guess to sum up my answer, I would have to say Clevelanders get mad out of envy, frustration and arrogance. It isn't that we don't want to see our neighbor and the capital city of our state to do well, it is just that we don't want Cleveland to be forgotten and not doing well.