Gust of Wind vs missile attacks

I don't see any inconsistency here. The exact language describing the effects of a strong wind is that it "extinguishes open flames and disperses fog." The language of Gust of Wind says "the gust disperses gas or vapor, and it extinguishes candles and similar unprotected flames in the area." So the Gust of Wind extinguishes unprotected flames, just like a strong wind would, and it "disperses... vapor" (which is what fog is), just like a strong wind would. Further, as has been pointed out, the specific language of Fog Cloud (as well as Stinking Cloud, Cloudkill, and Incendiary Cloud, to name a few) all suggest they are dispersed by a strong wind, "such as the one created by Gust of Wind." Gust of Wind even goes so far as to jeopardize protected sources of flame, which is something a normal strong wind does not do. A Gust of Wind is described as a strong wind in its spell block, as well as the spell block of at least four other spells, and it achieves the same effects as a traditional strong wind, with the added bonus of pushing targets in the area of effect, creating an obstacle to movement, and potentially extinguishing protected sources of light. It therefor stands to reason that it would do all the other things a strong wind does, including imposing disadvantage on ranged attack rolls with weapons, forcing flying creatures to land, and, in a desert, imposing disadvantage on Wisdom (Perception) checks.

 Strong wind in Gust of Wind is not capitalized, indicating it is not a term of art, so it should not be assumed to have any specific rules weight. This does not mean that a DM cannot reasonably allow it to do things that aren't specifically called out in the spell, D&D 5e is not a game of 'things do what they say and nothing else', but it would not be wrong for a DM to rule that it doesn't affect missiles.

Quote from Pantagruel666 >>

 Strong wind in Gust of Wind is not capitalized, indicating it is not a term of art, so it should not be assumed to have any specific rules weight. This does not mean that a DM cannot reasonably allow it to do things that aren't specifically called out in the spell, D&D 5e is not a game of 'things do what they say and nothing else', but it would not be wrong for a DM to rule that it doesn't affect missiles.

Unfortunately that is not a principle they adhere to when writing the rules so it doesn't help our determination. The "Strong Wind" rule in itself uses three different ways of capitalizing the term.

Strong Wind

Strong wind imposes Disadvantage on ranged attack rolls with weapons. It also extinguishes open flames and disperses fog. A flying creature in a strong wind must land at the end of its turn or fall.

A strong wind in a desert can create a sandstorm that imposes Disadvantage on Wisdom (Perception) checks.

Quote from Thezzaruz >>

Unfortunately that is not a principle they adhere to when writing the rules so it doesn't help our determination.

It's something they adhere to when referencing the rules glossary -- which does not include strong wind.  To be even more helpful, control weather lists five levels of wind, with 'strong' at level 3, and doesn't specifically reference the DMG definition either, nor does it give any definition at all for the other wind levels. Given the words it uses, it's plausible to link to real world scales:

1. Calm: Beaufort 0 (Calm->Gentle Breeze)
2. Moderate: Beaufort 4 (Moderate Breeze)
3. Strong: Beaufort 6 (Strong Breeze)
4. Gale: Beaufort 8 (Gale)
5. Storm: Beaufort 10 (Storm)

but this isn't really helpful, because while a strong breeze will force most birds to land, it won't particularly push humans around, that's more in the range of gale force and up.

It’s very confusing. I personally think it could go either way.