If you don't have savings for a down payment, and are renting, then there wouldn't be a difference. But if you've saved/invested money instead of putting it in a down payment, you could probably expect returns on that.
If you have a Wyoming LLC holding company (which are effectively anonymous when paying for a registered agent service) that owns & manages your in-state LLCs, then good luck finding you. This is generally how celebrities hide from stalkers, etc.
This is also beneficial when you're managing your own properties that you rent out. Tenants that know you're the owner will try to walk all over you. When you just tell them that you're the property manager, you're just enforcing the owner's wishes. You're "just the middleman". It's not their business to know that you're both. This also helps protect your separation of your business entity from yourself if for some reason they have to sue your LLC. If you don't represent yourself as the owner (because your LLC is), you have fewer problems to deal with.
Also, taxes like the other poster mentioned.
Plus if your life circumstances change and you want to turn your home into a rental, you're going to want to do this anyway and not having to sell your home to your LLC will save you a chunk of cash.
It can also be done for free using various methods, rather quickly too.
Also you don't have to put down 20% if you're willing to pay PMI and a bank is willing to lend to you, but to be competitive some banks are foregoing PMI (likely self-insuring the mortgage? Not sure how they can but I've seen it done)...which is craaaaazy to me.