There are arguments for and against both approaches. Briefly:
In favour of owning:
(1) may work out somewhat cheaper if you fly a lot;
(2) no problems scheduling (although you will still have to work around your instructor's schedule);
(3) probably easier to learn on one aircraft rather than bouncing around different 'planes that have different radio setups or whatever.
In favour of renting:
(1) no worries about not flying if the plane has maintenance problems (presumably an FBO would have several aircraft to choose from);
(2) allows you to focus on learning to fly rather than also having to learn how to purchase and look after an aircraft;
(3) most flight schools will charge about 10% more for instruction if you are using your own aircraft;
(4) may allow you to gain experience on two or three different types;
(5) doesn't require a lot of cash up front (pay as you go).
There are plenty of other pros and cons ... essentially, there is no 'right' answer that fits every student's situation.