Depends on what you really want.
HTPC - Everything is in one big box. And since we're talking Bluray and Streaming, sounds like you'll need some serious Hard Drive space to accomplish this, which takes up physical space. Multiple terabyte drives just to start. I assume this because having a 1,000 DVD collection in digital format, requires a ton of space. Bluray takes even more, so you'll need several 2TB or even 3TB drives to hold a big collection. Keep this in mind, depending on what you have going on. The HTPC will need to hold all this, or stream it from another device (but having several devices seems pointless when you are going HTPC as the point is to just have it all in one machine anyways). A HTPC is just a PC, but it's focus is to output audio & video. Generally doesn't need anything fancy, just needs to be quiet (ie, minimal noise, fans, etc), small (not always a requirement, this is based on your look/space), with plenty of hard drive space to hold your media (unless you're building a HTPC that just streams, but this is honestly kind of pointless, just get a streaming device for way cheaper). A robust HTPC can be powerful enough to handle all your media, rip the media, store the media, and play it all back. And you can of course change the front end software to your whim. That's the benefit. The con is that it takes up physical space, requires more maintenance (software updates, hardware failure, etc), and tends to get costly as you beef things up.
Streaming or Media Player - Basically a tiny little "computer" but without all the bells & whistles, with nothing robust, just a basic device that has specific hardware that can handle audio & video content. It holds nothing, it streams the data from an external location that is either local (via LAN, USB) or wireless (wifi, WAN, etc). These are inexpensive and there are tons of options out there right now. Usually uses things like APPS to play things like Netflix or other "streaming" websites. Also plays back your digital media (your ripped music & bluray/dvd collection). You need a big storage machine some where though. You can sometimes just have the hard drives with your media ripped to them attached to a streaming device via USB or LAN and you're good to go. Other times, you need it wireless perhaps and all your media is in a storage machine, such as a NAS, computer with file sharing enabled, or some other solution. The pro is that these are inexpensive devices and work well. The con is that they are not robust, they don't get updated much, and are outdated and replaced it seems every 6 months or so and sometimes have really clunky UI's and are sometimes slow or laggy.
So think about what you think you need based on your physical space, what your theater looks like (maybe you don't want to see a big computer box, or maybe it doesn't matter), and think about what would be the simplest and least expensive solution for your needs based on your collection size too (with room to grow).
After having done the HTPC approach, and having the streaming devices, I like both, but I actually use the streaming device with attached storage more for my movies instead of a HTPC. I wanted it simple with a remote and inexpensive. The HTPC always costs more, and controlling it requires a keyboard/mouse (bleh) or a fancy remote that just adds to cost. I like that the HTPC is easier to update, manage, alter, tweak, etc. But at the end of the day, when browsing for a movie to watch, I wanted it simple with a basic remote and nothing to configure/setup all the time, so I use my streaming device more (I use Boxee Box and a WD HD Live in two different rooms). I stream my media (1,000+ DVD's) from USB attached hard drives (external USB HDD enclosures with a few 2 TB drives full of movies). I also stream over my Wifi from shared folders from my main PC (which I use to rip/encode my media in the first place). Has a small foot print. And costed way less than building a full on HTPC from scratch.
I prefer my HTPC for things like Netflix, Amazon Prime, Hulu, YouTube, etc. I find streaming devices are just poor for these types of websites and are laggy or just clunky or just don't work properly for long.
I prefer my streaming devices for just playing back music and video that I've ripped to a hard drive, streamed over Wifi from another machine, or from locally attached USB drives.