The biggest difference between leasing and rental in my experience is that a lease is for a fixed period and rental is generally either for shorter periods or on a more or less day to day basis, for this reason rental will be somewhat more expensive.
My preference is to buy outright or if finance is the only option, to source my own finance.
In the 80s I worked as a copier salesman for a few months, the commissions we got for lease or rental deals was more than double the commission for an outright sale, it went up significantly if we could get the customer to sign up for maintenance/toner/paper deals too.
It seems to me the current fad for providing everything as a service is based on similar profitability.
Rental if you read all of the small print may be more flexible but it is essential that you know exactly what you are paying for.