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So there's three answers:
1. bandwidth
2. compression algorithm
3. color range
The first problem is color range. We use only 8-bits or 256 levels for each of red, green, and blue colors. This is enough when colors are far from each other, like bright white on top of black, but not enough when they are close, black on slightly darker black.
In the future, HDR TVs with 10 bit color will improve upon this, but in the meanwhile, we are at a level of poor color resolution where problems will show themselves.
The next problem is bitrate: the higher the bitrate, the better the video. Even if two videos support the same resolution, like 1080p, all else being equal, the video with the higher bit rate will have better quality.
Lastly, and probably the most important issue, is compression codec. Broadcast and cable TV uses old codecs which do worse for the same bitrate. They are built for people who don't change their TV or settop boxes for decades, and hence need to be backward compatible.
Services like HBOgo, though, can use the latest codecs. The software running on a PC can be updated, or even when using a TV with older hardware/software, it can still inform the servers the best possible codec that it supports.
"Broadcast" by definition sends the same stream to everyone, so can only support the least common denominator, the people with the oldest codecs.
A related issue is recompression. The video comes from HBO at the highest resolution with the best compression and is recompressed on its way to cable broadcast. Each time it's recompressed, it loses quality.
Back to bitrate. The bitrate from your cable box has fixed width channels with fairly low bitrate, something like 6mbps for 1080i hidef channels. Something like HBOgo can easily stream at 40mbps
Again, it negotiates. Some homes will see really high rates, whereas other homes will see things compressed down to 3mbps, and see quality worse than the cable channel, depending on Internet congestion in their area.
If you are getting high quality from HBOgo right now, I'll bet its because you've one the lottery and live in a neighborhood with lot congestion, because prime time (7pm to 10pm) is the most congested time for the Internet because online video.
So the answer to why the person is getting higher quality HBOgo than cable TV:
1. because the dark scenes make problems more apparent because low dynamic range
2. better bandwidth
3. better compression algorithms
Update: Also, this thread. The recompression cable companies use to transfer video in realtime have limitations that online codecs don't have. Among other things, online codecs can be more bursty, because more buffering.
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