Streaming only transfers the portion of the video you are about to view so there is drastically less bandwidth used, however, YouTube has opted to use progressive video instead.
Why does YouTube use progressive video instead of streaming?
(Obviously conserving bandwidth is not their top priority)
It really depends on what you're doing with video as to which choice you make, you may need to use both technologies to make the most efficient use of your bandwidth and streamline user experience. e.g. streaming may be more appropriate for longer videos, progressive for shorter ones.
The main advantage of progressive is that you only need a conventional web server like Apache, no specialised media servers, this has a huge impact on software and hardware cost across a network the size of youtube's and also makes entry level video distribution much cheaper.
Progressive video also decreases server setup complexity which has an impact on server uptime and maintenance costs (literally less to go wrong) and makes life much easier when creating or integrating with a Content Delivery Network because of the absence of specialised protocols. A few years ago you would have had to use a special streaming CDN, now standard CDN's can be used which reduces costs, increases scalability, and engenders a more competitive marketplace.
I also suspect it is easier to implement ABR (Adaptive Bitrate Encoding), which allows a video to be encoded and streamed while switching between multiple bitrates based on the users network bandwidth - much better for mobile.
There are pro's an cons to this e.g
In order to start the playback from a particular point of the video you will need to wait until it that point has been downloaded; if the video has a duration of 30minutes and you wish to start playback at min. 25, you will have to wait... wait until almost the whole file has been downloaded (until minute 25) in order to start the playback.
But a large network like youtube's can route round problems like this by using adaptive and progressive streaming intelligently.
