Ever is a Negative Polarity Item ('NPI'; see this recent post for explanation and behavior).
At all is also an NPI, but it's not temporal like ever, it's not a variant of any (like ever), and it has quite a different function.
At all is an NPI intensifier, referring to -- and emphasizing -- total negation of whatever is being negated, rather than referring to the extent of time when something could have happened, but didn't, like ever.
There is no logical difference between a negative with and without at all:
If he didn't show up for the party at all, then he didn't show up for the party; and vice versa.
I think (and here I'm guessing) that at all is a variant of the Minimal Direct Object NPI structure: eat a bite, drink a drop, give a damn, sleep a wink.
- If you don't eat a bite, then you don't eat anything, you don't eat at all
- If you don't even sleep a wink, then you don't sleep at all.
Etc. At all has some interesting syntactic properties; for one thing it can occur by itself, but it plays well with other NPIs (boldfaced) and frequently appears with them, as in this example:
- The candidates never mentioned it at all. (Neg trigger = never)
- The candidates never said anything about it at all.
- I'm surprised he showed up at all. (Neg trigger = surprised)
- I'm surprised he ever mentioned it at all.
- They kept her from leaving at all. (Neg trigger = keep from)
- They kept her from budging at all.