Do "periodically leave the house" and "periodically go out of the house" mean the same thing?

Question

A post says

As coronavirus lockdown restrictions ease and cities reopen around the world, the way you'll go to restaurants, malls and hair salons will change, so it's a good idea to think about how you'll keep yourself and your loved ones healthy during a prolonged period.

If you're not part of a vulnerable group, becoming a hermit isn't necessarily the solution, either. It's beneficial to periodically leave the house to maintain your health and sanity and run critical errands, but there are measures you can take when coming within spitting distance of people outside the household.

Does the following mean the same thing?

... "periodically go out of the house" ...

Do both of them mean "going out in public for a few hours"?

Is it grammatically correct and natural?

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leave the house or leave home? is related to this topic, though no one there talks about whether it sounds natural.

Answer 1

"Leave the house" is how you would describe departing to go somewhere else, such as going to work, for example:

I leave the house at 8am every morning to reach work by 8:30am.

"Go out of the house" could mean just to step outside of it - for example, into the garden, for example:

If you go out of the house you'll see the damage to the door.

There is an idiomatic use of "get out" which is often used to mean leaving the house to go elsewhere, for example:

It's nice to get out of the house for a while and go for a walk.