Does a streamer do anything to the data that Tidal provides?

Not sure what you want with the ethernet and power plug and why. The Chromecast can use an additonal/optional wired connector if your wifi is not strong enough, and it comes with a power plug (in fact the wired connection uses a special net adapter). What it will not do (unlike the Airport Express) is serve as an additional router to beef up your wifi network (you will need something else for that).
One other difference is that the Chromecast only uses your smartphone or tablet to extablish the connection and as a remote. The actual streaming is done directly from your router to the Chromecast. The downside is that if you want to give it a new command that may take a litle while (and not all apps are very good at this), but the advantage is that it does not need your phone/tablet to actually stream, so youir phone's battery is not drained and you may even turn it off.
Finally, the AE is only 16/44 while the CA goes up to 24/96 (if you care about this).

Many thanks for a lot of inspiring responses. I must admit, though, that I still do not understand, how a streamer can enhance the bitstream.
 
If the streamer is a transporter that connects the music sources with the DAC then how can it make the output any better than the input? 

I understand that if the input is provided in higher format than a streamer can manage, then a better streamer can do a better job. But if a cheap streamer can manage the input stream from say Tidal and provide it to the DAC bitperfect, then the job is done - isn't it? 

You are absolutely right. It cannot in any way improve the sound. If it is working properly what comes in will come out in exactly the same bitperfect shape and form.
Of course, if it also includes a DAC (quite a few do) that DAC may influence the sound. But the streamer part is neutral.

Many thanks, Willemj, 

So if my aim is to increase the sound quality of my setup (only listening to Tidal HiFi through Raumfeld streamer, Musical Fidelity DAC and Musical Fidelity amp) then I should start by getting a better DAC as the current streamer does 24 bit/192 kHz, right? 

I doubt a new DAC will give much better sound quality, if any. If you want better sound I would look at the speakers, as they are responsible for by far the largest part of sound quality (but I don't know what speakers you have). Do you have enough amplifier power (Musical Fidelity has many amplifiers)? What size is the room and does it have a hard acoustic, or not?