Other than one having a positive invariant scalar product and the other a negative one, what are the actual physical differences between these vectors?
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.
Although there are already three answers, they don't really answer anything. Each and every one of them fails to distinguish between vectors as part of a tangent space (e.g. four-velocity) and vectors which are present due to linear structure Minkowski possesses because it is flat. The second kind of vectors are obtained as differences of the space-time events. It's very dangerous to conflate these two notions and I daresay it's a source of some of the standard SR paradoxes. The second kind of vector structure is not present in general curved space-times and one instead has to talk about curves.
Now, it's not clear which type of vector you ask about. If only about the space-time event differences then this is basically what Vladimir talks about.
If you are instead interested in elements of the tangent space (to some event) then we are talking about notions such as four-velocity and four-momentum (for the purpose of this answer I won't distinguish cotangent space from tangent space) whose physical significance is entirely different from the above notions. The squared norm of four-velocity can be one of $-c^2, 0, c^2$ depending on whether it is time-like, light-like, or space-like. The corresponding squared norm of four-momentum is squared rest mass, which respectively splits into massive observers (with subluminal speed of propagation), massless observers (with luminal speed) and particles with imaginary mass known as tachyons that propagate superluminally.
In certain reference frames these intervals are just time (duration of a process) and a distance ( length of a still rod).