The "Attack action" is a specific action; attacks are anything with an attack roll.
Before learning the difference between a lower-case-a-attack and an upper-case-A-Attack this section on page 192 of the PHB is useful:
When you take your action on your turn, you can take one of the actions listed here, an action gained from your class or a special feature, or an action that you improvise
This "or" clearly means that class features and the "action listed here" are distinct actions. The first action listed is "Attack"; the "Attack action".
The most common action to take in combat is the Attack action,... With this action, you make one melee or ranged attack.
This, and the above quote, clearly show that the Attack action and attacks are different things. There is only one Attack action, which is why it is defined alongside the other actions such as Dodge, and Hide.
Lower-case-a-attacks are defined on page 194 of the PHB:
If there’s ever any question whether something you’re doing counts as an attack, the rule is simple: if you’re making an attack roll, you’re making an attack.
And so, bringing all this together: the Attack action is a very specific action that anybody can take whereas class granted features and abilities, such as the Ranger's Volley, are not Attack actions, but instead their own individual actions which may happen to involve attack rolls.
- On The Fighter's "Extra Attack" Feature (PHB 72):
Beginning at 5th level, you can attack twice, instead of once, whenever you take the Attack action on your turn....
This helps to showcase the difference between the two. If "attack" and "Attack action" were synonymous and thus if attacking meant taking the Attack action, then the Feature could simply read "You can attack twice, instead of once, on your turn."
This type of reading would also lead to allowing for infinite attacks in a turn, clearly not the intended reading.
This feature is not an action nor attack of its own, but a passive feature which activates "whenever you take the Attack action on your turn".
*The reason "Extra Attack" is capitalized is the same reason any feature or spell is capitalized, because they are a specific, proper, phrase/thing.
- On various features, such as Two-weapon Fighting (PHB 195):
When you take the Attack action and attack with a light melee weapon...
Again if "Attack action" and "attack" were synonymous then this sentence is redundant and could be reduced to "When you attack with a light melee weapon", the fact that it is not reduced further shows that they are two different things.
In order to cast most spells you must use your action to take the "Cast A Spell" action. If "attack" and "Attack action" were the same thing and you were casting, say, firebolt, a spell which has an attack roll, then you would have to spend your action on "Cast A Spell" but then have to take the "Attack action" as well, which is clearly impossible as you, generally, only have one action per turn.
Clearly something having an attack roll does not make it an "Attack action" or else a consequence would be the inability to cast spells with attack rolls.
- On Grappling/Shoving (PHB 195):
...you can use the Attack action to make a special melee attack, a grapple....
Using the Attack action, you can make a special melee attack to shove a creature,...
These are attacks made using the Attack action and are specifically called out in the rules as being "special". They are special because they can replace the Attack action, and not simply any attack. The best example of this is Opportunity Attacks, which cannot be altered into a grapple or a shove as you are not taking the Attack action but a Reaction that requires an attack.
- On capitalization and wording:
As Nitsua60 correctly points out:
"A close reading of the PHB makes clear this is a place where the authors have been very consistent and explicit in differentiating: you'll always see phrases like "make an attack" and "take the Attack action," never "make an Attack" or "take the attack action." See PHB pp. 49, 63, 72, 74, 78, 79, 85, 92, 93, 111, 165, 168, 170, 192, 195, for some examples of their consistent and exacting capitalization."
Even from a grammatical standpoint it is clear that "Attack action" and "attack" must be different things. It is not standard English to randomly capitalize nouns in the middle of sentences, unless they are proper nouns, which is the case here.
Again, "Attack" is a specific action; attacks are anything with an attack roll.