Calanctus:
To answer your question without too much technical detail:
1. A speaker that is time-accurate means that the drivers are arranged in a spatial relationship to the listener so the sound they produce arrives at the listener's ear at the same time. This can be done by mounting the drivers on a baffle that slopes away from the listener, or by tilting the speaker away from the listener. The objective is to ensure that the audio signals from each of the drivers hit the listener's ear simultaneously, as is true of live sound.
2. A speaker that is phase-accurate means that there is very little or no difference in the phase relationship between the audio input signal from the amp, and the movement of the driver(s). You are probably familiar with speaker phasing in regard to the connection of speaker wires: if you reverse the positive and negative leads of one speaker (relative to the other speaker), then the drivers of one speaker will be moving forward while the drivers of the other speaker are moving backward. This leads to acoustic cancellation because the phases of the two signals are opposed.
In the context here, being "phase-accurate" has a somewhat different connotation. A speaker that is phase-accurate uses a first-order crossover, which results in the drivers responding in the same, or nearly same, phase as the input signal. Crossovers of the second, third, fourth, etc., order shift the phase relationship, so the speaker movement may be 45 degrees to 180 degrees out of phase with the input signal.
There are a number of positive aspects to using a first-order crossover in a speaker, but there are some design problems that must be overcome:
1. first-order crossovers are usually of lower efficiency than higher-order crossovers;
2. the components in a good first-order crossover, such as the resistors and capacitors, must be of high quality;
3. the drivers used with first-order crossovers must be of high quality and closely matched to ensure flat frequency response.
There are very few speakers that are both time- and phase-accurate. Three of the best known are Dunlavy, Thiel, and Vandersteen, and each has a distinctive "family" sound. (Actually, I think most speakers that are made by companies headed by someone who is pursuing a personal vision of speaker design -- as in the case of John Dunlavy, Jim Thiel, Richard Vandersteen, and Jacques Mahul of JM Labs -- tend to have an identifiable family "sound".)
Our man Sean is the technical guru in this discussion, so I will let him add any clarifying remarks.