If you can afford it, it is the best way to be taught, being taught in your own plane, by a CFI who will help you learn all the little things about your plane and your technique is extremely valuable. It is priceless.
You can practice on soft field landings, because it is your plane. Most flight schools will not let you use their planes, for soft field landings because of insurance.
On your cross country flight, you can plan and take a real cross country, because cost will not factor in your decesion and you will try to plan a real trip. Your CFI will help you plan the trip and make it safer for when you do get your ppl. For example, Like going to fly to visit your sister in another city. You can even spend the night, something most flight schools will not let you do with their student planes. I believe the more you learn the way you will fly with the help of your CFI the safer you will be after you get your PPL. Statistically speaking, your most dangerous time as being a pilot, is the first several hundred hours after you get your PPL.
You can use your plane whenever you have free time to fly (after approval of your CFI). Thus you will have many more flights closer together (which is what we all want). You will start flying it, the way you will fly after you get PPL. The advantage is that you will still be on the tutelage of your CFI for all sorts of little questions along the way, that is worth it all by itself.
You will also learn all the important maintenance questions that will be on the ppl exam, because they make sense, your have real personal interest in the FAA plane/owner requirements.
If you buy a used plane, the value of the plane will not depreciate much, and you can sell it for more or less for what you paid for it. You will also learn all about the fixed cost of plane ownership. It is definitely not the cheapest way of learning, but the most complete way of learning if you want to be a GA.
You are much more likely to finish your PPL if you own your own plane.
It is expensive, but you generally get what you pay for.
On a side note:
It is controversial, but I would much rather be taught on a complex, retractible plane and learn it first that way, then to learn on a fixed propeller, fixed gear plane and then try to transition to a complex plane. I believe "gear up landing" are more likely in pilots that learn on a fixed gear plane first, but that is just my opinion. it is easier to learn good habits.