Ultimately it's a matter of cost (hardware vs. integrated/windows). I've worked with various levels of consumer to enterprise RAID for many years, but I still consider myself intermediate so someone more knowledgeable is welcome to chime in and correct anything incorrect, but here is a thousand foot level:
Windows Raid : I would not even consider this real RAID. You take a small performance hit (roughly 75%-80% of the speed) vs. using IRST, and it's labeled as "Spanned", "Striped", and "Mirrored" by the OS instead of the usual Raid0/1/5/6/etc. Has limitations on sizes and harsh requirements for MBR/GPT configurations to optimize.
Intel or Marvell RAID : IRST is actually pretty sound. Configurable via BIOS or with the IRST software, allows for the typical important things like e-mail alerts (you'll have to run your own mail server relay, usually), and has average performance if you pair your system with a UPS and enable write-back cache (vs. write-through without). I personally avoid Marvell integrated RAID like the plague due to extremely poor performance (worse by far than Windows "RAID"), but some of their actual hardware solutions are alright.
Hardware RAID : Dedicated hardware controller doing the processing (vs. your CPU handling the RAID in Windows/Integrated mentioned prior) which leads to the fastest performance without noticeable system impact. Either an integrated card for servers or commonly a PCI-E (1x/4x/8x) card for desktops and HTPC's. Has the highest flexibility for number of drives. Options include built-in cache for insane speed boosts to throughput and battery backups connected to the controller in the event of power-failure. Downside is, obviously, cost - with low-end cards starting around $300 and high end anywhere from $800-$2000.
Generally speaking using the integrated RAID chip on a motherboard makes the most practical sense for things like HTPC's and home-use, purely because the controller can cost as much as all the rest of your equipment combined in these situations. I don't do storage as my day-to-day, so this is just my two cents as well.