The biggest advantage of cash is being able to buy a distressed property on the cheap. Banks know that cash offers are more likely to close. As a result, you can get a property for offering less than someone else who needs to get a mortgage. Some houses do not qualify for a mortgage, drastically reducing the buyers pool. After you pay cash, then you can decide to take out a mortgage or not.
Deciding on a mortgage probably depends on your personal goals. People who wish to acquire lots of properties will put the least amount of money down. If you do not wish to acquire multiple properties then don't take out a mortgage. If you think you might want to get one more then consider a 15-year mortgage.
For me personally, if I could get a 15 year at 3.125%, which is the current rate at my credit union, it would be difficult for me to pass that up. I would do it. Even a 30 year at 4% seems like a great hedge against inflation.