I've never heard of people feeding seaweed instead of regular horse feed. But then again, in Arabia they used to feed them milk & dates, in India it was often spiced vegetables, even meat. And in the western world, we tend to feed buckets of cereal grain & other unsuitably high carb feeds...
As others have said, it is very high in iodine. It has a wide range of 'trace minerals' too, which is why some years ago people got excited about feeding it as a natural 'general nutritional supp'. But it is so low in everything else compared to iodine, I'd just feed it AS an iodine supp if need be. For eg. when I used a program(FeedXL.com) to balance my horse's diets years ago, they came out great in everything aside from iodine with the 'ration balancer' I was giving them. So I just added a sprinkle of seaweed meal to their feeds for that purpose. Currently though, I am using a powdered supp that has seaweed in it already, so they don't get extra.
The bad thing about iodine is deficiency symptoms are the same as toxicity symptoms. Iodine content in seaweed can be very variable and without testing you just don't know what level
There is a fair bit of leeway there though, and if they're in 'the range' it should be fine. That goes for most nutrients, as does the need for them to balance with others - eg. it's not enough to just provide 'adequate' Ca ~ it needs to be correctly balanced with P & with Mg or there will be health issues. But some nutrients are quite 'touchy', such as Selenium, and have smaller 'safe range' and can be disasterous in deficiency or excess.
Deficiency or excess symptoms - and problems - are also commonly the same or similar for other nutrient balances too. Eg. If you have too much Ca & very low Mg, you will still have weak, crumbly bones.