The primary reason for studying Lie algebras is the following fundamental fact: the representation theory of a Lie algebra is the same as the representation theory of the corresponding connected, simply connected Lie group.
Of course, the representation theory of a Lie group in general is very complicated. First of all, it might not be connected. Then the Lie algebra cannot tell the difference between the whole group and the connected component of the identity. This connected component is normal, and the quotient is discrete. So to understand the representation theory of the whole group requires, at the very least, knowing the representation theory of that discrete quotient. Even in the compact case, this would require knowing the representation theory of finite groups. For a given finite group, the characters know everything, but of course the classification problem in general is completely intractable.
The other thing that can go wrong is that even a connected group need not be simply connected. Any connected Lie group is a group quotient of a connected, simply connected group with the same Lie algebra, where the kernel is a discrete central subgroup of the connected, simply connected guy. So the representation theory of the quotient is the same as the representations of the simply connected group for which this central discrete acts trivially.
There is a complete classification of connected compact groups. You start with the steps above: classifying the disconnected groups is intractable, but a connected one is a quotient of a connected simply connected one. This simply connected group is compact iff the corresponding Lie algebra is semisimple; otherwise it has some abelian parts. In general, any compact group is a quotient by a central finite subgroup of a direct product: torus times (connected simply connected) semisimple. Torus actions are easy, and the representation theory of semisimples is classified as well. Whether your representation descends to the quotient I'm not entirely sure I know how to read off of the character. When the group is semisimple (no torus part), I do: finite-dimensional representations are determined by their highest weights (which can be read from the character), which all lie in the "weight" lattice; quotients of the simply-connected semisimple correspond exactly to lattices between the weight lattice and the "root" lattice, and you can just check that your character/weights are in the sublattice.
All of this should be explained well in your favorite Lie theory textbook.
Here is a brief answer: Lie groups provide a way to express the concept of a continuous family of symmetries for geometric objects. Most, if not all, of differential geometry centers around this. By differentiating the Lie group action, you get a Lie algebra action, which is a linearization of the group action. As a linear object, a Lie algebra is often a lot easier to work with than working directly with the corresponding Lie group.
Whenever you do different kinds of differential geometry (Riemannian, Kahler, symplectic, etc.), there is always a Lie group and algebra lurking around either explicitly or implicitly.
It is possible to learn each particular specific geometry and work with the specific Lie group and algebra without learning anything about the general theory. However, it can be extremely useful to know the general theory and find common techniques that apply to different types of geometric structures.
Moreover, the general theory of Lie groups and algebras leads to a rich assortment of important explicit examples of geometric objects.
I consider Lie groups and algebras to be near or at the center of the mathematical universe and among the most important and useful mathematical objects I know. As far as I can tell, they play central roles in most other fields of mathematics and not just differential geometry.
ADDED: I have to say that I understand why this question needed to be asked. I don't think we introduce Lie groups and algebras properly to our students. They are missing from most if not all of the basic courses. Except for the orthogonal and possibly the unitary group, they are not mentioned much in differential geometry courses. They are too often introduced to students in a separate Lie group and algebra course, where everything is discussed too abstractly and too isolated from other subjects for my taste.
Best Answer
Neither statement requires compactness as a hypothesis. The key result to both, and the only place where any work is needed, is the following:
Both statements you want are exercises assuming this result.
For the connection between representations of a simply connected Lie group and its Lie algebra, take $H = \text{GL}_n(\mathbb{C})$ and use the fact that complexification is left adjoint to the forgetful functor from complex to real Lie algebras.
For the connection between Lie groups having the same Lie algebra, if $G$ is a simply connected Lie group and $H$ is a connected Lie group with the same Lie algebra, then there is an isomorphism $\mathfrak{g} \sim \mathfrak{h}$, and by the above it lifts to a map $G \to H$ which is a local diffeomorphism and hence (with a bit of work) a covering map. It's an exercise to show that a covering map between connected topological groups is a quotient by a discrete subgroup of the center. Since $G \to H$ is surjective, a representation of $H$ is determined by the corresponding representation of $G$, and the only condition to check is that it factors.
The key result should be in any book that discusses the relationship between Lie groups and Lie algebras; to give two random examples, it is proven in Section 8.3 of Fulton and Harris, and it is Theorem 3.27 in Warner's Foundations of Differentiable Manifolds and Lie Groups.