Worth checking the source
The Register's link was broken, but after a few web searches I found the article.
The little picture in the Register's article compares 3.28 hours of streaming with 17.28 hours of DVD (50% mail delivered, 50% consumer delivered). That 50/50 split was a guess because the authors did not have any better numbers. The full sized oranges to oranges picture with separate columns for mail and consumer delivery is here.
By the study's numbers, streaming is only more environmentally friendly when consumer transport is included. Mail delivery is about as environmentally (un)friendly as streaming. If you use consumer delivery, you can divide the big pink rectangle if you watch a rental more than once, or if you rent more than one film per journey (I think the study assumes pick up and drop off in the same round trip journey of 17km). You can also squash or remove the pink box if you went to town for other reasons and bought more than one DVD while you were there. The idea of buying a boxed set of six for a TV series has been conveniently forgotten.
The next biggest box is consumer equipment power. Old DVD players use the same amount of power when idle and when playing, and these make up a significant fraction of the 2011 US population. If I buy a second hand DVD player and leave it on continuously, then for every Joule it uses, 0.99 Joules are not used for central heating (the house is too hot for about 4 days per year). The same would be true of a streaming box, but in the study, streaming boxes use about half the energy of DVD players.
This leaves streaming with a great big green box for network transport energy that make it tower over the tiny grey and yellow boxes. Much of the big green box would be there anyway for an internet connection whether you use streaming or not. (The invisible brown sliver for mail delivery would still be there because the post lady delivers letters too.) The only box that survives is the grey box for the embodied energy (manufacturing costs) of the consumer devices. This is about the same for streaming and DVD.
The real conclusion should be: don't waste time worrying about the tiny speck of energy used by video. Concentrate of the things that matter: transport, central heating/cooling, and washing. Little things like switching to an efficient fridge and fluorescent lighting save you money, but even if everybody does those little things, then end result is a little change in the total amount of power used.