The fact that only connected Feynman diagrams contribute to the scattering amplitude can be interpreted in terms of the vacuum of the theory. Omitting disconnected diagrams amounts to a shift of the vacuum: the vacuum of the interacting theory differs from that of the free theory.
Regarding your second question: strongly connected (also called one-particle irreducible) diagrams are needed in order to calculate loop corrections to the propagator. The exact propagator is given by a geometric series consisting of one-particle irreducible diagrams. Furthermore, they play a role in the calculation of the exact vertex function.
I can recommend two excellent and free sources for more information on the subject: David Tong's lectures on QFT and Mark Srednicki's book.
This post imported from StackExchange Physics at 2014-03-24 04:02 (UCT), posted by SE-user Frederic Brünner