Beginner here. Need help on understanding the difference between the below set up. Why is that blender give me two different finish when I do 'double value' vs 'double texture node'?
The results are different simply because your two examples are telling Blender to perform completely different things - the Noise Texture cannot be split in that way to produce the same result.
In your first example you’re using the input coordinates (effectively Generated coordinates) to generate noise at a scale of 34 and roughness of 1. In your second example you’re using the input coordinates (the same as in the first example) to produce noise at a different scale but then using the output of that (which then won’t relate to the original coordinates) at yet another scale (the Scale represents the scale of the texture in texture space, not the amount of noise as you seem to assume).
This second example is effectively producing noise that varies based on the amount of noise from the first Noise, rather than on the position in the texture.