Fifth edition uses natural language, and these phrases are certainly different when viewed through that lens. A weapon that counts as magical for the purposes of overcoming resistances is not magical.
First let's just look over what the two phrases even are:
a magical weapon
That would be, a weapon which is magical. If anything requires a magical (or non-magical) weapon, it will count as a magical weapon. Compare this to the following wording:
a weapon that counts as magical for the purpose of overcoming resistance and immunity to non-magical attacks and damage
This weapon is non-magical, but when it is being used to overcome resistances it counts as magical (A small thing that is unclear and matters more later is whether the weapon actually briefly becomes magical). If anything else requires a magical (or non-magical) weapon, it will count as a non-magical weapon.
This pretty clearly showcases that these are different things and for further evidence that these phrases mean different things: they both exist. Why would both phrases exist if they were equivalent; the books could simply have used one phrase every time, but instead both phrases occur regularly.
Now let's look at how this distinction ends up mattering throughout the game, first let's look at things which only work on non-magical weapons such as the animate objects spell:
[...] Choose up to ten non-magical objects within range that are not being worn or carried [...]
This spell would not work on magical weapons, but it would work on weapons that count as magical only for overcoming resistances.1
Now let's look at the Warlock's Pact of the Blade which has a feature that only works on magical weapons:
[...] You can transform one magic weapon into your pact weapon by performing a special ritual while you hold the weapon [...]
This would not work with a weapon which merely counts as magical for the purposes of overcoming resistances, but will work with any weapon that is actually magical.
There are ways to make a weapon flatly magical, for example the shillelagh spell which states:
[...] The weapon also becomes magical, if it isn't already [...]
This would work perfectly on either type of weapon here, especially since it includes the phrase "if it isn't already".
There are also ways to make a weapon that merely counts as magical for overcoming resistances such as the Warlock's Pact of the Blade feature which states:
[...] This weapon counts as magical for the purpose of overcoming resistance and immunity to non-magical attacks and damage [...]
1 Note, this creates a problem/question: Do non-magical weapons of this sort actually become magical when overcoming relevant resistances or merely count as magical? Imagine that such a weapon is animated and attacks something resistant to non-magical damage: If the weapon actually becomes magical then it would no longer be a valid target of the animate objects spell, which is a bit of a problem. This is a separate question for sure.