Most good 24bit recorders cannot achieve more than a 20bit or so noise floor thanks to the limitations of analog components.  With the best equipment you can get a noise floor about 30-32dB lower than the limitations of 16 bit. That is theoretically 30dB of free digital gain!  If the end format is a CD, that is.   
 
Most ambient noise levels are also above a 16bit noise floor, so in real use, there isn't much benefit to anything higher.   Still, it is good practice to record in 24bit and give yourself some headroom to work with, but not too much, mind you .  
 
I record in 48kHz 98% of the time since a lot of my recordings are accompanied with video, and I'm too lazy to switch sample rates the rest of the time.  Like drewfx1 said, a higher sample rate with allow you to record higher frequencies, full stop.  There is no other practical use during recording....seriously.  All else is fiction, speculation, and conjecture.  On a side note, the distortion of most microphones increase exponentially once you pass 20kHz, even ones designed to reach 30 or 40kHz.  And if you feed that to a speaker system designed for 20kHz playback, you may also increase the distortion throughout the entire range of the tweeter.  All bad things!!! so I would recommend using the maximum sample rate for the needs of the material you are recording.  

Then there is PonoMusic. This apparently does not get any thoughts on these types of Forums. Recording and archiving your sounds of songs on the highest rates possible may prove fruitful in the future even as many are not currently being on board with the PonoMusic concept.

Going back to the OP's comment that they thought the exported WAV was not the same quality/did not sound the same as the song played through the DAW ----- to test this could this be an option :
 
- Export the song as a WAV at 44.1/24bit (ie, the project setting)
- Import that WAV back into the project and invert the phase of that track
- If its an exact copy it should cancel out the sound of the project playing and you hear silence
 
* I'm not certain I got these steps/or logic correct --- if someone can confirm?

SimpleManZ
Then there is PonoMusic. This apparently does not get any thoughts on these types of Forums. Recording and archiving your sounds of songs on the highest rates possible may prove fruitful in the future even as many are not currently being on board with the PonoMusic concept.

The problem with the PonoMusic concept is that the masses using portable digital music players are used to hearing the compressed sound of an MP3. And they don't care about higher resolutions in sound. And no matter what any of us say they're not going to change their mind since to them the MP3 sounds good...

The problem with Pono is it's BS.
 
Higher sample rates just allow for higher frequencies that aren't present in music at a volume that is audible to humans.
 
Higher bit depth just means less noise, but in the overwhelming majority of listening situations the quantization is already buried under other noise at 16 bits.
 
The overwhelming number of people - including many, if not most, audio professionals - really have no idea how loud the 16 bit quantization error + dither is being played back at under typical listening conditions. Thus it's easy to convince them that "16 bits aren't enough" because they have absolutely no clue how many bits they need to ensure that quantization error + dither isn't audible.
 
 
EDIT: Oh, and since people are often clueless about human perception and lazy too, they don't bother to do careful double blind tests and then convince themselves in lazy comparisons between CD and "hi res" that might in fact be different masterings with different EQ played back at different levels (which obviously will be audibly different) and/or they believe they hear magical differences that have nothing to do with "less noise" or "higher frequencies present".