I'm a programmer and I love rockets. I don't work in Aerospace and all of this is from my own knowledge and learning. Take that as you will.
The SSMEs (Space Shuttle Main Engines - Aerojet Rocketdyne RS-25), required an almost complete rebuild after every launch. It was nominally reusable in that the parts could be reused, but it took a long time to refurbish and re-test the engines. Also, the first stage of the Space shuttle included the solid rocket boosters and the hydrogen fuel tank. The boosters were absolutely reusable (I seem to remember reading that of all the SRBs, all but four were reused - two from the first launch were never recovered, and two from the Challenger disaster that couldn't be recovered). But the Space Shuttle wasn't fully reusable like the Falcon first stage.
To me, the difference between the Space Shuttle and the Falcon 9 is kind of like the difference between a doctor of mechanical engineering and a mechanical engineer. The Space Shuttle was an amazing (and complex) piece of tech - the engines used liquid hydrogen and liquid oxygen, which is incredibly efficient but notoriously difficult to work with (cooling hydrogen to a liquid sate and keeping it cooled is not trivial). It was a long (~10 years) process to develop the Space Shuttle from the existing tech we already had. The result was a highly tuned and complex engine that was incredibly efficient. It makes you feel a bit warm and fuzzy at how efficient and well engineered the engines are.
SpaceX went the other way - they said we'll use a slightly less efficient set up (RP-1 (refined kerosene) and liquid oxygen) as our fuel, but we'll make up for it by making each individual launch much much cheaper. So while they may not have the fuel efficiency of the Space Shuttle, for the cost of a single Space Shuttle launch (~$400M per launch according to wikipedia), SpaceX could launch 6 or 7 Falcon 9s (~$60M per launch). The SSMEs are very efficient engines. The Falcon 9 is a very efficient platform. It's kind of like a big picture vs micro optimization sort of thing. (There is no judgement here - the Space Shuttle was developed by a government with much different concerns and constraints than the private business developing the Falcon 9).
Having a fixed wing aircraft come down from space really doesn't make that much sense when we get down to it. It's a really great and interesting concept, but the realities of it make it not worth the trouble. Sure the Space Shuttle glided, but even a brick going 17,000 miles an hour will glide for a long while. This video (https://www.youtube.com/watch?v=B3JZtY7bcbM) gives a great overview into the complexities of a plane-shaped spacecraft hitting the atmosphere at 7 km/s. When a capsule (like the Apollo Command Module) hits the atmosphere, you only need to protect one side of it, and you just let the thing fall like a stone then slow it down with parachutes. The Space Shuttle Orbiter had to have different materials with different heat shielding properties all over its body because it would glide through the air at different angles. The bottom of the orbiter had an enormous surface area, and it needed to be thermally insulated. But heat shields are very heavy, and that amount of weight would make the launches impractical and landing impossible. So NASA had to develop new, light materials that were incredibly strong and also heat resistant.
The Falcon 9's first stage is not going nearly as fast as the Space Shuttle Orbiter when it hits the atmosphere, so it's not really appropriate to make a direct comparison about the complexities there. But this is to me another part of the brilliance of SpaceX looking at the big picture - it's much easier to reuse the first stage (which is typically the most expensive part of a launch vehicle) which doesn't reach orbital velocity and capture as big a chunk as the savings as possible rather than being totally 100% efficient.
I would not be surprised if NASA was 50 or 60 years ahead of its time with the Space Shuttle. It seems like private industry is catching up to where NASA was with the Saturn V in the 60s just now (we're still not close to a rocket as powerful as the Saturn V). We may see that in 20 or 30 years, the industry uses the materials and knowledge gained from the Space Shuttle to develop new and novel spacecraft.
Again, I'm just a programmer who loves rocketry and space - I apologize if there are any technically inaccurate bits in this post.