The first statement establishes that bases exist, and the second statement says more:
Not only do bases exist, but you have the freedom to extend any linearly independent set to a basis.
Obviously if you can extend any linearly independent set to a basis (you coudld just start with a single nonzero element, for example) then bases exist, as given in the first statement.
Another example of this would be
In a ring with identity, maximal ideals exist
In a ring with identity $R$, given any ideal $I\neq R$, there is a maximal ideal of $R$ containing $I$.
The second statement is clearly stronger.
There are situations where the first sort of thing holds, but not the second. For example, projective modules always have maximal submodules, but I believe I saw a post here once that showed you could not always find a maximal submodule containing a given submodule.