What is U-duality? and Why U-duality is important?

As you mentioned, string theory compactifications have some symmetries. By definition, the U-duality group of a string compactification is the minimal and non trivial group that contain the T and S duality groups in the sense that the former is not a direct product of the latter groups. Relevant reference: U-duality and M-Theory

U-dualities are cool. For example, they can be mixures of $T$ and $S$ dualities, that means that is sometimes possible to stablish dictionaries not just between space parameters like $R rightarrow frac{1}{R}$ or between couplings as $e^{phi}rightarrow e^{-phi}$ but to perform transformations that exchange spacetime moduli with couplings or viceversa.

Now notice that by definition there must exist $U$-dualities that are not products of $T$ and $S$ dualities (otherwise the $U$ duality group could be written as a product). New, wonderful and strongly stringy physics comes into play by considering those transformations. I strongly encourage anyone interested in string theory to read the following fabolous papers: Exotic Branes in String Theory, Exotic Branes in M-Theory and the blog post Exotic Branes, U-branes and U-folds.

Other consequences: The jungle of three dimensional supersymmetric field theories is wide wild (see Cordova's talk What's new with Q?). Why are three dimensional supersymmetric theories vastly more difficult that its higher dimensional counterparts? At least for those ones derived from string theory the answer is that truly exceptional $U$-duality groups arise after a dimensional reduction at (and below) three dimensions.

It is very likely that a correct physical understanding of U-dualities would seed light to answer many of the most fundamental questions in theoretical physics, namely: What is the physical principle undelying string theory? or how a unified version of string theory (covariant under U-dualities) really looks like.