If you live in an apartment, you have to pay a rent to someone.
If you rent an apartment, it is someone else, an owner.
If you bought an apartment, it is yourself, namely, you will pay to yourself!
From the viewpoint of finance, there is no differences.
However, buying an apartment is margin trading of a real estate. You may well be in debt more than your yearly income. Your financial portfolio would be heavily leaned to real estate.
My financial policy is "Don't put all your eggs in one basket."
I rather want to rent an apartment and buy stocks and REIT (real estate investment trust). If I had more than one million dollars, I would afford buying an apartment around $250,000.