Unfortunately, this is not just a common misunderstanding/error but so common it’s almost ubiquitous and indeed, much of the audiophile world relies on it! Namely:
You change one or more components in your setup, say a “better cable”, DAC, etc., and you hear a difference or even a surprising difference, sometimes even when you don’t expect to. It’s the most basic logic/“common sense” to conclude that the change you made must be the cause of the subsequent difference you hear. We learn and experience such “cause and effect” almost from the moment we’re born to the moment we die and it’s so obvious we hardly ever or never even think about it. However, this “common sense”/logic/reasoning can be fooled, almost always because there’s some other variable (other than the obvious change we’ve made) that has also changed, which we’re unaware of or haven’t considered and is the real cause of the effect (difference we hear).
Almost always, there are a number of these “other variables” and audiophiles typically either aren’t aware or don’t consider any of them, or consider them briefly and erroneously discard them as the cause. In this case, the most likely variable you haven’t considered is that you’re actually comparing different masters (as
@bigshot mentioned) but there are potentially others, such as different loudness normalisation between the two services and the ever present perception biases.
Your link to spotify’s “audio quality” is very informative, apart from the fact it doesn’t give any indication at all of how that audio quality relates to audible differences in quality. With most material it should be easy to hear a difference with 24kbps but by around 160kbps it’s virtually always audibly transparent, let alone at 256kbps VBR. So this variable (256VBR vs Flac) cannot be the cause of audible differences, no matter how good your reproduction equipment/environment.
G