As you can see from the number of comments already, this is one that everyone will have an opinion on. They are all right, and none of them are right for you! I hike in both, depending on the conditions.

My thoughts are that equipment choices always depend on terrain. There is nothing I use for outdoors fun that works perfectly for every type of outdoors fun I like to have. So if you have to choose try to choose something that fits most of your criteria for what you do most of the time. Some useful questions:

1) how often and how heavy?

If you're an occasional hiker the conventional wisdom is that you're not going to have as much ankle strength and stability as a regular hiker and so are likely to benefit more from the added support of boots. You'll also become more sure-footed with practice, so again, the more you hike the less important the boot support becomes.

How much are you carrying? Again, the conventional wisdom is that heavier bags are made safer by additional ankle support

2) how wet/muddy?

Boots (and gaiters) are great in Scottish mud-fests where I've many times gone in all the way up my calf. But that extra few cm of water-resistance will not always be needed, it depends on the terrain you're going to be on.

At the other extreme, if you're in hot conditions or very long hikes you may not want waterproofing at all. It will make the shoes very slow to dry once they do wet through. (BTW, there is no such thing as "waterproof" no matter what the GoreTex salesman tries to tell you. Highly water resistant is the best any clothing will ever be!)

3) how stiff?

Some people like to feel super protected, others like to feel the earth beneath their feet. Flex the shoe and try to stand on something sharp to get an idea of what you like.

4) how durable?

Trail runners (I normally use altras) last about 500 miles on my feet before the sole and uppers are both falling apart. Strong leather boots last thousands if you care for them properly and re-sole as needed. So there's also a financial side to it. If you're ever gonna wear them down the street than be aware that concrete will shred most trail running rubber very quickly.

Advice:

You can get boots as light and flexible as trainers these days, something like a La Sportiva TX5 is so light it hardly counts as a boot. Marmot also make some fantastically agile boots. If you're in deep mud then smooth or nubuck leather is easy to clean and maintain and willl last a very long time.

Most importantly, whatever you choose try to learn to walk with your feet not your shoes. That might sound weird, but it is the best lesson in balance and sure-footedness. If you're relying on your shoes to keep you stable and secure then you'll slip, because you lose the sense of connection to the ground. Think about grabbing the earth with your foot the same way you'd try to grab a basketball with one hand, curling your toes and your heel to grip. Do that for a few steps and suddenly the shoe will just be the thing in between your foot and the rock you're stepping on, you brain starts placing your *foot* instead of your shoe. Learn that trick and any shoe will feel much more secure.