Other than one having a positive invariant scalar product and the other a negative one, what are the actual physical differences between these vectors?
Answer
A timelike vector connects two events that are causally connected, that is the second event is in the light cone of the first event. A spacelike vector connects two events that are causally disconnected, that is the second event is outside the light cone of the first event.
In that sense, the timelike vector can be considered to define a four-velocity direction of an observer and thus the time axis of that observer (to be a four-velocity it should be normalised). On the other hand a spacelike vector can be considered as defining a space axis (a spatial direction) of an observer. Seen like that the two vectors define a time interval or a length on the appropriate inertial frame, as Vladimir said.
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