Firstly, movement by shuddering or shivering, in simplicity, effects the conversion of potential energy into kinetic energy. This is achieved by expending ATP in muscular tissue, causing rapid bursts of contraction.
Brown adipose tissue, although located within most mammals ranging from infants to elderly, is predominantly found in infants differing from white adipocytes insofar as brown adipose tissue has a greater capillary network. Due to the higher surface area to volume ratio of the infant comparative to adults, the rate of heat loss is appreciably greater than a human adult. Thus a more efficient means of thermogenesis is required, namely, when the BAT detects body temperature below the homeostatic norm, the hormone norepinephrine is released, reacting with lipases to form triglyceride. This may subsequently decomposed further, to be converted into heat energy around the body (there are of course various intermediate stages).
Additionally, located primarily in brown adipose tissue is a protein named UCP-1. This allows the cells respectively to undergo 'non-shivering thermogenesis', heat generation without shivering by uncoupling the respiratory cycle in the mitochondria to oxidise more rapidly the substrate. As a consequence, fewer molecules of ATP are produced but heat production is increased.
Therefore, the differences may be rested on 1) the required efficiency of heat generation between infants and adults; 2) the act of shivering per unit of muscle area generates more heat.
You can find further information regarding the UCP-1 uncoupling mechanism in BAT from, https://www.ncbi.nlm.nih.gov/gene?Db=gene&Cmd=ShowDetailView&TermToSearch=7350