I bought my tank after researching & calculating lease vs. Buy. Tanks need a hydro test every 7 years, which costs $25. Generally you hand in your old tank & get another tank back full. You still own a tank, but what tank that's changes. I've had my tank filled & returned occasionally too though.
I hear some places don't like customer owned tanks & will not fill them or not fill them without ownership documentation. Haven't ran into that or any issues showing up with just a tank & no paperwork myself at the couple local welding shops I use here in Colorado. They are more mom & pop types though. Big guys like Airgas might be more picky.
You don't get hit with hydro test charges if you lease, but $25 every 7 years is a lot cheaper than a $10-30 annual lease.
I just upgraded from my 150cf? tank to a 250cf? tank if I recall. Cost $70 & apparently happens a few times a day at my local welding shop with people needing more capacity or more portability.