First of all, signal strengths are all relative.
Small signal analysis is the measurement/calculation of the tangent of the transfer function. But large signal is actually moving along the transfer function.
When a circuit is perfectly linear, all signal is small since the tangent is constant. However, almost all active components will have some kind of non-linearity, which makes transfer function non-linear. Therefore, large signal analysis is to examine the behavior of these non-linearity, or worse, functionality.