There is a risk of confusion here because in some sense Minkowski space is too nice. What I mean by this is that in the setting of Minkowski space, because of its simple structure, there are identifications and globalizations possible that do not make sense in a general spacetime.
In a general spacetime at every point you can define the tangent vector space. It roughly has one direction for every direction you can move in. This does not mean that the space is an affine space (an affine space is like a vector space without a preferred origin), it could be a sphere for example. But for Minkowski space, the spacetime is indeed an affine space and this leads you confusion of the whole space with the tangent space.
Let us talk about relativity. An observer in spacetime can find three spacelike curves and one timelike curve through his or her time and position. The principle of Lorentz invariance is that any choice is fine! For the space part this is just that you can rotate your laboratory and get the same results. That you are allowed to mix time and space comes from that the speed of light should be the same for observers in relative motion.
So really, what local Lorentz invariance means is that you can rotate your laboratory without changed results, and observers moving relative to it see the same physics. This is an expression of symmetry in the tangent space.
Now in Minkowski spacetime pick an arbitrary origin. Then Minkowski spacetime has the same structure as the (1+3) tangent spaces of general relativity, so the local Lorentz invariance can be made global. Since the origin was arbitrary you have also four translation symmetries. This is the Poincare symmetry.
Local Lorentz invariance is a statement about how your local choice of time and space axis is unimportant. Global Lorentz and Poincare invariance is a much stronger statement about the symmetries of spacetime itself. In particular, a spacetime need not have any symmetries at all (and there are many known examples of solutions to Einstein's equations that don't).