In general, larger drivers can play lower (very gross generalization here). And, there is an advantage when trying to achieve optimal imaging to having a broader range of frequencies coming from a single source. Ideally you want to cover the entire vocal range from one location (big tweet, fullrange, or tweet+mid) if you can.
On the other hand, the point at which a driver "beams" is directly related to effective cone diameter. The beaming point is the frequency limit above which the driver becomes more directional and off-axis response starts to roll-off. (Again, in general) larger drivers will start to beam at lower frequency, so positioning of a larger tweeter can be more critical. That said, sometimes off-axis mounting can help 'tame' harsh highs by leverage the natural roll-off above the beaming point.
In addition, by lowering the cross-over point between midbass and a larger tweeter you are avoiding more of the beaming frequencies in the midbass driver - which tends to be problematic when they are mounted lower in the doors. To me this is the biggest advantage of tweeters that can play low when using a typical two-way component setup. You can keep the midbass in their omni-directional range and bring more frequencies up to ear level and create better imaging.
Of course, things are a little different for cone-based, dome-based and planar drivers.
(EDITs were made for clarity)