Why are finite-dimensional simple Lie groups so special? Because they admit a full classification on the level of (complex) finite-dimensional simple Lie algebras by combinatorial data, e.g., root systems and Dynkin diagrams. Semisimple Lie algebras are then just direct Lie algebra sums of these simple Lie algebras.
By Weyl's Theorem, finite-dimensional linear representations of semisimple Lie algebras are completely reducible, i.e., a direct sum of irreducible representations. Again the irreducible representations of simple and semisimple Lie algebras can be classified by combinatorial data, e.g., using highest weight theory.
Nothing of this sort is true for solvable and nilpotent Lie groups. We can again pass to the level of Lie algebras, but a classification is impossible in general. Even for complex nilpotent Lie algebras, a complete classification is only known up to dimension $7$.
The representation theory of solvable Lie algebras is also very different. Over the complex numbers, every irreducible representation of a solvable Lie algebra is $1$-dimensional by Lie's Theorem, but a result as Weyl's Theorem does not hold. So there is no way to classify linear representations of nilpotent and solvable Lie algebras and Lie groups.
There are many references in the literature. The wikipedia article on Semisimple Lie Algebras has further references, as well as the article on its Representation Theory. Finally, there is the nice article on Simple Lie groups. This has applications to geometry, e.g., to Riemannian symmetric spaces.