In geometric terms, the Lie algebra is the tangent space at the identity element of the Lie group. Since a tangent space is flat, or linear, that obviously makes it easier to work with than the Lie group.
The tangent line to a function at a point on its graph tells you how the function is behaving in the neighbourhood of that point, but it doesn't tell you anything about the global behaviour of the function. In the same way, the Lie algebra tells you what the Lie group looks like in the neighbourhood of the identity. By group multiplication you can translate the identity to any other point of the group. So the Lie algebra gives information about the behaviour of the group in the neighbourhood of any point. But it cannot tell you anything about global properties such as the topological structure of the group (for example, whether it is connected or simply-connected).