No they aren't. An attack roll is an attempt to do damage, a saving throw is an attempt to reduce or avoid damage.
Something that resolves an attack to do damage IS THE SAME THING as something that resolves an attempt to avoid taking damage from an attack. Both are a roll of a die vs a difficulty class (armor class or saving throw number) to determine the exact same thing, which is "does he take damage"?
It's literally only a difference between whose hand is rolling the die. The mechanic is identical, and the purpose is identical.
Just because you roll a D20 for both of them doesn't make them functionally identical. For that to be true, they have to have the same result, which they don't.-
They do - either you take damage, or you do not take damage. Same result being decided by both. Fair warning, this response of mine is a trap for you.
Until you put in DOAM, then a hammer is the functional equivalent of a Fireball, which makes absolutely no sense. I can't imagine anyone would ever try to make the claim that a hammer and a fireball are functionally identical.
It's funny, when people want to talk about the difference between the two, they often use fireball. But that's not a good example as it doesn't encompass nearly the quantity of things covered by the concept - in fact it happens to be the least objectionable thing covered, which is no coincidence I suspect.
No it also covers a straight-line attack like a lightning bolt (why does it do half-damage if you dodge it?), a spell which requires an attack roll like Fire Seeds, and a weapon that does splash damage like Alchemist Fire. All of those raise serious issues. For example, evasion eliminates all damage on a save even with no cover in a narrow corridor while the guy with the best dex and reflex save on the planet still takes at least half damage even if he has nearby cover (this makes no sense and isn't a corner case - it comes up in most parties where the rogue has evasion but no cover to dodge behind but someone else with a good dex has something to dodge behind but still takes half damage). Alchemist Fire damages someone with a force field in effect that would prevent even the most devastating blow from a deity (this makes no sense - why do first level minor cheap splash weapons pierce all protection?). And spells like Fire Seeds requires an attack roll, just like Great Weapon Fighter (but nobody ever objected to it all these years).
This combination of factors which are all rolled up into the "acceptable" damage on a miss mechanics from prior editions makes the whole concept far more questionable than merely saying "fireball".
So no, the hammer is not equivalent to fireball. But it is equivalent to some of these other far more sketchy things that are included under the larger umbrella of "acceptable damage on a miss mechanics" that you were fine with all these years, and that's where this debate really focuses, and not on the easy example of fireball.