Reading over some of the posts again, I think some of the confusion is coming from a terminology problem. When people in or associated with the industry talk about mechanics or mechanical impact they generally understand the term to have a particular meaning. This is good, because it allows us to have conversations without having to define terms. However, it also means that if someone has a different understanding there will be confusion.
We all know that cosmetics effect the experience. Purchasing players get enjoyment from them; it allows them to express individuality, or to stand out, or to show that they're supporting the game. It makes them happy. This is why they purchase.
What cosmetics don't do is make the player deal additional damage, or reduce miss-chance, or increase their speed. This is what is generally understood to be a mechanical impact.
Let's look at a quick hypothetical.
In AwesomeShooter, Ahab Whaleson purchases some cosmetics because he wants to stand out; he gets a cool overcoat and sea-captain's hat. Standing out makes Ahab happier, so he enjoys his gaming experience more. Moby Dickson also plays AwesomeShooter but doesn't like microtransactions; maybe she can't afford them, or maybe she can but just doesn't think they're worthwhile. Moby isn't disadvantaged by Ahab's purchase because there is no mechanical impact.
Now a competing game called AmazingShooter gets popular. Ahab Whaleson purchases an awesome harpoon instead of the standard weapon. It looks awesome, and it deals +10 damage, so Ahab is happy. This time, Moby is unhappy because Ahab's purchase gives a mechanical advantage. Suddenly the game seems unfair. Moby complains about "pay to win", enjoys the game less, and may even stop playing.
This is the difference between cosmetics and mechanical purchases that everyone else has been talking about.
Now, to be fair, in the real world, even well implemented cosmetics may have a negative impact on the experience for some non-purchasing players. Some might feel jealous if they can't afford the items, or may feel it's unfair if they can't earn them through gameplay. Experience has shown us these are a minority - otherwise retention would be damaged and it would no longer be profitable. Some might feel it ruins the feel or experience of the game. Of those, some will just enjoy the game less, while a smaller subset might stop playing or will start vocally objecting. Again, experience has shown us that this is a small enough number of players that it's worthwhile to implement cosmetics. We can't make everyone happy.
Does that clarify what people mean by the difference between cosmetic and mechanical purchases?
Quick disclaimer: for simplicity, I'm talking about well designed and implemented cosmetics. We're all aware that sometimes a dark- coloured skin in a dark environment might in fact impart a slight mechanical advantage for example, but as that's an implementation issue rather than an inherent issue with cosmetics it's probably not worth getting bogged down on it.