Fair question.
I think that the major difference comes down in the case of a data breach.
When it will happen, even if you created aliases like:
They will still be linked to your account that will probably have the following:
- predict@proton.me (or your more
<real name attached to it>@proton.me)
- your bank details, with your real name attached to it
- IP logs of where you connected (or even your personal house address)
- etc etc
Hence enough to cross-reference all of your aliases to you.
On the opposite side, if you buy anonymously, youâll pretty much have the following:
- some random username or long string of numbers that will probably be very non-personal
- some aliases blending into the mass to provide decent anonymity
So, at worst when Addy will have a data breach, people will know that somebody somewhere used some aliases some day. Which isâŚquite vague and useless in itself.
Now, if the following are breached at the same time:
- Addy, to find out the aliases somebody created
- several companies you bought from, to narrow it down to letâs say
Mr kissuâs persona thanks to purchasing habits
- your main mailbox, where weâll see that kissuâs real name is actually Bob based on some emails that might have some mandatory personal info
Then, people will indeed have the full picture of it indeed.
I might be wrong and missed a few blind spots but if you actually do have most of those events happening at the same time (which is very unlikely), then you would indeed be screwed over. 
I mean, itâs still miles better than just putting your personal/work-related Gmail address in there with your real name on it. 
UPDATE: while writing this, I also double-checked some stuff on the side.
Addy does not allow a forgotten alias to be re-used, hence you cannot have several people using an alias you previously used.
Are you sure you want to forget xxx.yyy@addy.to? Forgetting an alias will disassociate it from your account.
Note: This alias uses a shared domain so it can never be restored or used again so make sure you are certain. Once forgotten, xxx.yyy@addy.to will reject any emails sent to it.
I guess itâs a good thing because in the end, if your main emailbox or a public-facing website is breached, you can always act upon in time and roll your email address with a new address.
Hence youâll need both Addy + (company or main mailbox) to be breached at the same time, which is not perfect but definitely better than nothing.
PS: I think I got it right, feel free to say if Iâm wrong or challenge all of it for me to find a flaw in my statement. 
Overall, it all comes down to: itâs hard to still be anonymous if all databases are breached all the time at the same time. 