If I had a mortgage somewhere I couldn't live like I do now when I can freely move to another country for a year for a gig, then move again. So renting gives you flexibility to move after jobs quite easily and travel the world while you're young, gain international experience and see other cultures.
Of course, once I settle down in one place where I'd like to stay I will buy a property instead. But I haven't decided where I want to live yet.
Once you buy a house somewhere you pretty much have to stay in that area for next 20 years and are limited to jobs in your local area. Of course, you can sell, but 1) that's a hassle and takes time 2) there might be a down market when selling for good price is difficult and then you're stuck in your place until market turns around. If you're renting you can move whereever economy is booming or there is some interesting niche job.
Long story short, if you rent, relocating for jobs or expatriating to a different country is easy. So I'd recommend renting while you're very young (<30-35) and buy a property later once you are more settled down.
For example, when I bought my DSLR, I looked at various camera rental sites and used prices of year old models. It showed me that I should buy the camera because even if I sell it after couple of years, my overall cost of camera would be only couple of weekends rent.
So generally I buy cars, books, video games, furniture, music CD (yes still buy CDs) etc. Selling is really easy. (Lately, books have no resale value but there is no book rental service other than library. Also I like to gift books so it evens out.)
Only thing I rent because of above math is movies.
Then there are a few things I rent just because it is convenient to rent or cost too much to own for short time period. So I pay for gym because I don't want gym equipment at home. Or rent cars when traveling.
For some (myself included) there's more "cognitive load" in owning stuff. I have to keep track of:
* Where is it?
* Where do I store it?
* How do I keep it safe?
* What upkeep does it need?
* When was the last time I used it?
* Do I really need it?
For personal use:
I prefer to rent things I don't intend to reuse indefinitely (movies, for instance). But I'm very careful about recurring subscription fees. They add up quickly. Netflix is my one non-utility subscription - mostly because it costs so much less than a cable TV subscription while offering such a better experience.
I prefer to purchase anything that might inhibit my access to data I intend to use in the long term (basically, any software for creating/editing non-ephemeral data). Also, a lot of software is ridiculously overpriced at rental prices.
I rent my apartment because purchasing an equivalent home would cost multiples of my rent every month - I basically couldn't afford it. Also, real estate is already overpriced where I live and thus a poor investment, I don't want the responsibility for upkeep, and I don't intend to stay in this location for more than a few years.
I rent cars because I only need one once or twice a year. I don't even take cabs more than about once a month.
Basically everything else I buy. Or don't need and thus don't buy OR rent.
For business, cash flow and spending money to make money flip some of the considerations on their head.
Renting software for collaboration makes more sense than self hosting as long as the costs are less than the value of the time you'd spend maintaining it. Other software is worth renting or buying (whatever the business model the producers use) simply because you need its features and you don't get a choice of how to pay. The increased productivity of the right package over the wrong one for an employee's core function is easily worth hundreds, often thousands per year.
Office space: rental is a no-brainer unless you're Apple or Google.
Furniture: if you expect your company to exist for more than a few years, buy. Furniture lasts a long time.
Computers: Buy Macs, rent Windows. Your engineers and designers need the latest and greatest once every couple years, sure. But the rest of the company will do fine with their hand me downs. However, the build quality on windows computers is a race to the bottom, and I've seen too many laptops fall apart within 18 months. Whereas probably 80% of Mac laptops are just fine 5 years in, if a little cosmetically blemished.
Appliances: rent. Rental companies seem to have fewer qualms about replacing your appliance and taking the old one back to repair, instead of stringing you along scheduling repairman visit after repairman visit. And you just don't want to be without a coffee machine or dishwasher for three weeks.