There's a secrecy reason as well for some (the FAA registry will show "N1234, LLC" as the owner) as well as a sales tax avoidance reason for others. ("I didn't buy an airplane for which I'd owe sales tax on; I bought a company for which no sales tax is due.")
In the case of liability avoidance, it's hard to do if the (real-world human) owner of the airplane is also the pilot at the time of any accident. They might not be able to sue you as the owner, but they can still sue you/your estate as the pilot, or the person who oversaw the maintenance, etc.
Many people where I am own GA aircraft as a syndicate - everyone is a shareholder in a company and you get x hours per year of flight time per share, and have to pay your share of the costs.
I imagine that the limited liability helps if there are say 20 people in the syndicate and you own a share of it. One of the other members can crash the aircraft into something expensive, and as long as it had nothing to do with you or your negligence, you shouldn't be liable beyond your share value going to 0 (or having to pay up any unpaid capital to the book value of your share)
An unincorporated partnership would make this much riskier.
In the case of liability avoidance, it's hard to do if the (real-world human) owner of the airplane is also the pilot at the time of any accident. They might not be able to sue you as the owner, but they can still sue you/your estate as the pilot, or the person who oversaw the maintenance, etc.