Hoof Boots and Shoes: What Each One Actually Does
They are frequently discussed as rival philosophies. They are better understood as two tools that solve overlapping problems in different ways, with different failure modes.

Ask about hoof protection and you will often get an argument rather than an answer, because the topic has acquired camps. It is more useful to look at what each option physically does, and where each one tends to go wrong.
What a shoe does
A shoe is fixed to the hoof wall and stays there through the shoeing cycle. That gives it a particular set of properties.
It protects the wall from wear, which is its oldest and most basic function. A horse working regularly on abrasive surfaces can wear horn away faster than it grows, and a shoe stops that.
It is always on. There is nothing to fit before work, nothing to forget, and nothing to lose in a field, which for a horse in daily work is a genuine practical advantage.
It can be shaped and positioned to influence how the foot loads and breaks over, which is why remedial and corrective work is mostly done with shoes. That is a farriery decision rather than a purchase decision.
The costs are real too. Nails go into the hoof wall. The horse is committed to that shoe until the farrier returns, and a shoe that is lost partway through the cycle frequently takes some wall with it.
What a boot does
A boot is removable, which is the entire difference and the source of both its advantages and its problems.
It protects the whole foot, including the sole, and it comes off afterwards. That suits a horse that is comfortable barefoot most of the time and only needs protection for particular work, which describes a lot of leisure horses.
Nothing is attached to the hoof wall, so there are no nail holes and no reliance on wall quality to hold anything, which matters for a horse with poor or damaged horn.
The costs are equally specific. Fit is critical and unforgiving, because a boot that moves rubs, and a boot that rubs stops being usable very quickly. They can be lost in deep going. They take time to put on before every ride, which sounds trivial and is the reason a lot of boots end up unused at the back of a tack room.
A shoe fails by coming off unexpectedly. A boot fails by not fitting, or by not being put on.
The measuring point people get wrong
Boots are sized to the hoof, and the hoof changes size through the trim cycle. A foot measured just before a trim is a different foot from the same one just after.
Measure immediately after a trim, and expect the fit to feel different at the end of the cycle. Getting this wrong is the single most common reason a boot that seemed right turns out not to be.
How to actually decide
The variables that matter are the horse's workload, the surfaces it works on, the quality of its horn, and its conformation. A horse doing daily road work on hard ground has different needs from one hacking twice a week on grass.
Which is why this decision belongs in a conversation with your farrier, and with your vet where there is any lameness or foot pathology in the picture. They are looking at the actual foot in front of them, and that beats any general principle in an article.
The useful thing an owner brings to that conversation is honest information about how the horse is actually used, rather than how it might be used in an ideal week.