If you go to page 9 you will see Cuyahoga County's Conventions and Visitor Bureau numbers. They are netting a revenue of about $8.5M more than Akron per year and that's after contributing ~$1.5M to Gateway, $232K to the convention center, and $1.4M to the Rock HoF. They had a surplus of about $4.7M for 2015 after accounting for all expenses, A number that I imagine will skyrocket for 2016 considering the number of hotels they opened and the rooms they had filled for the RNC.
Long story short Cuyahoga County has a huge surplus in bed tax money to spend so spending it on the venue that can bring it large conventions, like the RNC, makes sense. Summit County pretty much has their bed tax eaten up via overhead. Salaries, property insurance, utilities, food services, repair and maintenance, & Supplies eats up 75%+ of what they collect. That doesn't really leave them with any room to finance any large scale arena (or anything for that matter). Cuyahoga County doesn't need to put a new tax on the ballot to fund an arena because it already has revenue generating from a preexisting one to fund it. Summit County would need a new tax as it doesn't have anywhere to pull that money from.
http://ohioauditor.gov/auditsearch/Reports/2016/CVB_of_Greater_Cleveland_15-Cuyahoga.pdf