I don't understand the difference. How is a melee weapon attack different than a melee attack with a weapon?
The 5th edition game divides up all attack rolls with two categories and two types, so that any attack roll made should be one, and only one, of the following four things: melee weapon attack, ranged weapon attack, melee spell attack, ranged spell attack.
It then separately classifies weapons based on which section of the weapons chart they appear in, so that any given weapon is one, and only one, of the following: melee weapon, ranged weapon. This is why weapons that appear on the melee weapon section of the chart and have the thrown tag are counted as melee weapons even when they are used to make ranged weapon attacks (which only matters because it sets which features can be applied, such as that a thrown melee weapon can benefit from the dueling fighting style's +2 damage, but cannot benefit from the archery fighting style's +2 to hit).
And lastly, the game (post errata) says "Instead of using a weapon to make a melee weapon attack, you can use an unarmed strike: a punch, kick, head-butt, or
similar forceful blow (none of which count as weapons)." which creates a situation in which a melee weapon attack is made with a thing that is not classified as any type of weapon (the errata even removes unarmed strike from the weapons chart to make that bit clear). Improvised weapons are similarly not weapons, but can be used to make melee weapon attacks and ranged weapon attacks.
So, to summarize, "melee weapon attack" refers to the type of attack roll being made, but not the weapon or non-weapon being used, while "melee attack with a weapon" refers both to the type of attack roll being made, and that a weapon must be used.