So Weapons change your health stats, as does attire
In this image you can see that by equipping the War Sword over the Two Handed Sword Noctis gains both Health and Attack.
In this one we can see that the change in Attire (going to No Jacket) Increases he Attack, his strength, and what I can only assume is Penetration. But trades those increases off with a decrease in Health.
This is what I though would happen, as only being able to change your stats through Accessories seemed a bit minimal. (Just as a note, the change in his attire drops his health by 77, from 466 down to 389, which isn't minimal, but certainly not major)
What I didn't expect is that the weapons equipped would change your stats. This leads to so very interesting combinations. Imagine if you really liked only using the Engine Sword, and maybe some daggers. This means your other two Slots can be used to simply raise the stats you want with no consideration of using the weapons. So if you really hate the lances, but you found one that increases you Strength by a large amount. It's going to make you Engine Sword better, but you don't have to use it. - This would be cool, but it's only paritally true. Check below.
Edit: Quoted the images because they're massive.
Edit 2: Ok, so I was watching the video back again because I wanted to know how the magic affects those overall stats, and I realised that the weapons aren't adding to overall strength, and that the arrows illustrating what goes up and down are specific to the weapons and how they compare to what's being equiped over them. What I did find interesting is that they still had an overall affect on Health. The only things I could figure out that was making the Attack stat go up was their attire, and so in any accessories as well.
Be warned that this image is 5760x1080
I used Gladios in my example because the limited amount of things he can equip makes it easier to illustrate. The same things apply to Noctis and the Magic doesn't have an affect on the strength. When you see the player equping the Magic, you can all see that weapons affect the amount of MP, and for some reason the spell they equiped gave less MP than the Daggers ><