Capitalization is a red herring: the real distinction is between "attack" and "Attack action".
Both of the answers from nitsua60 and Medix2 are correct regarding the nature of the distinction. However, the distinction is between the phrases "Attack action" and the word "attack," not between capitalizations.
For completeness, an Attack action is a type of action that can be taken, defined on PHB 192:
The most common action to take in combat is the Attack action, whether you are swinging a sword, firing an arrow from a bow, or brawling with your fists...
Certain features, such as the Extra Attack feature of the fighter, allow you to make more than one attack with this action.
Whereas an attack is defined as anything that requires an attack roll (PHB 194):
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.
An Attack action can contain an attack, but it can also contain multiple attacks as well as grapples and shoves, among other things. In accordance with its name, using an Attack action uses up your action for the turn. On the other hand, an attack is usually contained within some other attempted thing: a spell, a melee weapon strike, etc.
Whenever the text is referring to an action, it is explicitly called an Attack action. Capitalization is not always a reliable way to determine whether or not the text is referring to the action--instead, one has to focus on the context. Attack actions are always written as such, whereas the other kind of attack frequently carries a modifier with it, such as "melee weapon attack" or "spell attack" or such. I have not found an exception to this rule in the PHB.
This is most commonly clear in the MM, where monsters' melee weapon attacks are all capitalized as "Melee Weapon Attack". Clearly these attacks are not attack actions, but they are capitalized as part of the header. If we focused only on the capitalization, we would have to look to context to determine whether it's an Attack action or just an attack.
I realize that this difference might be splitting hairs, but it's useful for resources that are not as careful as the PHB. For example, the official rules in the roll20 compendium or other SRD websites are not always so careful to preserve the capitalizations, but they usually do a good job in preserving the proper wording. Focusing on capitalization rather than terms of art might lead to frequent confusion.