Assuming a purely private flying operation, the principal advantages of owning an aircraft through a limited company are as follows:-
(1) Corporate ownership of the aircraft protects group members in respect of the civil liability which arises under section 76(2) of the Civil Aviation Act 1982 if the aircraft or something which falls from it causes damage to something or someone on the surface (eg by crashing into someone's roof). The point here is that this liability arises without proof of negligence. By contrast, liability for damaging something or someone in the air (eg by crashing into another aircraft, or injuring one's passenger) depends on proof of negligence.
Note that corporate ownership of an aircraft does not alter an individual's personal liability for his or her own negligent flying. Note also that, whether or not the aircraft is owned by a company, prudence dictates that it be insured in respect of liability to third parties.
(2) The ease of keeping a record of and of transferring shares in the aircraft.
As dublinpilot has pointed out, a company is one method of giving structure to what might otherwise be a somewhat vague arrangement. There are other methods, but a company is a fairly simple one.