Both BR and U leads to the condition of "not knowing", which limits the ability of agents to make fully informed or rational decisions which are needed for them to fulfill their objectives of maximising value for themselves. In this aspect, they are similar.
Technically, BR in a certain sense can be controlled if we have unlimited processing power and access to information. Information is needed to reduce BR, but too much information requires additional processing and hence may lead to increased BR.
Uncertainty, on the other hand, cannot be eliminated regardless of effort and time spent on reducing uncertainty. We simply CANNOT predict what might happen in the future. While we say that you are going to have the Ebiz exam on 5 May 2018, that day may not come if there is a sudden war or disaster that renders logistics impractical. This is uncertainty.
Take for example, a navigation device. The device can tell us what is the "shortest route" to go from point A to point B to reduce our BR, but it cannot tell us that there will be congestion just as we get started on our journey. So the device does not reduce uncertainty. However, if the device is supplemented by real time traffic data, the data can help to mitigate our uncertainty if it is timely enough for us to adjust our strategy. Hence, the more real-time the data, the more it is able to reduce uncertainty.
The same reason holds for supply chain management. In order to reduce demand uncertainty, companies do forward vertical integration to get closer to customers and get first hand demand information. This helps to tame the bullwhip effect.