Belts, aprons, pouches and bags are a very personal choice and are often dictated by the job at hand. Some jobs can be handled with what you can carry in your pockets, other times your entire inventory is at hand and you still need to go buy another tool.
That apron may very well work for you in most instances, however some of the drawbacks that I see to it are the pockets aren't big enough to get your hand into, to retrieve small items from the bottom (nails and screws). Where do you carry larger items like tape measure, torpedo level? Again this is dictated by the job you are doing whether you need these items.
I could see the apron being more useful in a shop setting. Certainly if you already have a belt, I wouldn't throw it out. Keep your options open.