"Real" streamer vs. Bluetooth adapter--what's the advantage?

Sorry - Spotify/Airplay isn’t a relevant comparison to Idagio/Amazon/Bluetooth. I shouldn’t have mentioned. 

Get a Bluesound node 2i.
it does Bluetooth, airplay and it’s own but I don’t know what it’s called.
see if you can tell the difference.

coys21’s description is accurate. Transmitting music with Bluetooth sends the music through the Bluetooth transmitter in the phone to the Bluetooth receiver that's part of your system. Bluetooth transmission compresses the signal from its original resolution, whether the music is stored on the phone or retrieved from a music service. There have been advances in Bluetooth quality but it started as a technology to connect keyboards and mice to computers and wasn’t originally designed for music.

When you instead use WiFi from the phone to a "real" streamer, the phone is just a remote control, as coys21 said. The phone tells a music service like Tidal, Amazon HD, etc., to send the music, uncompressed, through your modem and router directly to the receiver at the original resolution provided by the music service.

Since I would be using a phone app to tell the streamer what to play, why would the sound quality improve?

This is the same as using your TV remote to change channels or adjust the volume. The actual picture and sound quality is handled by the TV, the remote has no bearing on it. 

Get a Bluesound Node 2i, but make sure you DO NOT play via the bluetooth interface, or even Apple Airplay. Using direct ethernet or wifi  with the Bluesound remote app will give you the sound quality that will be substantially better than the other two options.

One of the beauties of the BS Node is that it allows so much experimentation on both the input and output sides. You can feed it by Bluetooth or Airplay, or direct.
I wouldn’t worry about using Airplay, if using it suits you. Airplay is lossless and serves up 44/16 red book - and nobody has been able to prove that higher rates or resolutions give audible differences.
Bluetooth on the Node is the aptX HD standard so if your phone is up to it this is “Hi Res” 24 bit. So actually “better” than Airplay (and standard CD) if you believe you can hear difference above 44/16.