Okay, here is my crack at it.
Devices:
All of these devices (Apple TV, Roku, Chromecast & certain TVs, DVD players, Gaming systems, etc) have streaming capabilities.
Apple TV, Roku, and Chromecast are dedicated devices whereas the DVD players also, obviously, can play DVDs and game systems can play games, etc.
IMO, the dedicated devices work better. I've tried a few TVs with clunky interfaces (Sony and Samsung in particular), two or three DVD players, XBOX360, PS3, and Wii.
I have no experience with Chromecast, so I can't speak to it well.
Channels
As far as content, each device has a number of 'channels' that it supports. Those channels hold the content. Some channels are free for the devices, some are not. Some are available on all streamers, some are not.
For instance, to the best of my knowledge, Hulu and Netflix are available on just about every streaming device you could find. HBOGo is widely available, too.
Other services like
Amazon Prime, and provider specific channels like Cinemax, Showtime, A&E, etc are not available on all devices.
If a network does not have their own streaming 'channel', they will often sell their content to another streaming channel provider - typically Hulu or Amazon for current TV shows.
So, in order to determine which device will work best for you, you need to figure out which streaming channel provides the content you are interested in, then find the streaming device that supports those 'channels'.
For the most part, Roku has a wider range than Apple TV. Roku has a ton of 'channels' - most are moderately useful at best, but they do have things like Amazon and A&E that I know of.
Content:
So, the next logical question is - How do I know what channel has my favorite shows? I use
http://www.canistream.it/. Takes a bit of clicking around to drill down to what you want, but it's a good resource.
Once you figure out which of those channels have your favorite content, then you need to figure out which streamer supports that channel. Google is probably your best bet there.
Cut cable?
Lastly, as someone said above, I don't think any of these will really let you 'cut the cable cord' unless you are willing to stop watching some things that you currently watch. You can probably find most of the 'big 4' network programs (you can do those over-the-air as well), but the 'smaller' networks (like the ones you mentioned) may not be provided anywhere else.
I have both Roku and AppleTV devices with subscriptions to Hulu, Amazon, and Netflix in the family, and we still end up watching a lot of things through cable (either DVRd, VOD, or Live).
I hope I didn't make it more confusing - ask more questions if you need more help.