It’s really personal preference. I prefer hiking boots over cross trainers; I’ve never liked the way low top shoes feel on my feet. That doesn’t mean boots are better, they’re just what I like to use. The only real advantage I’ve found is that full leather boots offer more foot protection. I can’t tell you how many times I’ve had a stick or pointed rock stab the top of my foot as I hike. The leather offers some impact resistance to this.
As for the cost of boots, for a beginner it really doesn’t make sense to pay top dollar for boots when a less expensive pair will work just as well. As one continue to hike more and more it become more cost effective to buy a more expensive boot. For example, I normally can only get a maximum of one season hiking season (I’m excluding winter from this as I where different boots that time of year) out of a $70 pair of boots, whereas I can get 4 seasons out of a $200 pair. Something to keep in mind is not all $200 boots are equal, nor are all $70 boots, so you still have to pay attention, but I’ve found that generally with boots, you get what you pay for. And always keep your eye on sales, sometimes you can find good pair of boots for much less than what they are normally priced for.