Yes and no. Your booking will entitle you to the same benefits regardless of where you make it. However, some agencies/TAs don't know/deal with WDW that often, so may not understand that they need to give you the client the WDW confirmation number to make your ADR's and plan your FP+. The Joes Travel Agency confirmation number they give you is worthless for stuff like that. So you can do it, but you may need to pester the agent to give you the correct information.
Typically prices are going to be the same across the board. Tickets may be cheaper through a 3rd party reseller such as Undercover Tourist or AAA, but thats usually maybe $20-30 savings on a $400+ ticket. Rooms are usually the same price, but occasionally Expedia or Priceline or similar will have a room listed. Dining plan is going to be the same price as I've never seen a formal discount on that (exceptions being some buy 5 get 2 free or Free Dining promos). Flights could be all over the place depending on how hard a particular TA/company is looking and if they are selling their own services (IE Southwest selling a package trip). There is only so much wiggle room in any of the categories. Say 10% fees paid by Disney to the TA when they place the booking. Now a particular TA may choose to pass some of that fee onto you as savings, or Southwest may choose to not make money on the hotel since they make more on your flight (as an example). If prices are off by more than 5-10% I'd check MouseSavers to see if there is perhaps a discount code available that you aren't using. With all that said, there is a chance that a tour operator got a great deal by agreeing to charter the whole flight, and book a huge block of rooms through Disney. But for a custom trip, I'd be a bit leery of a huge discount.