As a single driver fan, I'm not much of a crossover expert, but any crossover plays both drivers at or near the crossover frequency to maintain a flat frequency response. It's just a wider band of frequencies with lower order crossovers that have a slower slope and so a greater frequency range of overlap between drivers. Yes, you'll hear sound from two different locations, but I believe it's the phasing between drivers that is critical (many crossovers put the driver out of phase with each other). And any time you have two drivers playing at the same frequency you'll have comb effect (interference between two sound sources). Time alignment refers to physical (typically vertical) offset between driver acoustic centers (usually taken as center of voice coil). Note that two drivers can only be time aligned in one axis, the vertical axis allows for better horizontal imaging. Comb/alignment effects will vary with listener location and frequency. Note also that coaxial drivers take care of comb and time alignment issues.
As non-perfect transducers each driver will sound different (even matching ones on "2.5" designs), but most certainly between cone woofers and ribbon tweeters for example. A few speaker manufacturers use very high order, very sophisticated crossover designs to try to eliminate this phenominom but then must battle against the sound suddenly jumping between drivers as frequency slowly increases/decreases. One advantage of higher order crossovers is that the drivers only have to perform within a smaller frequency range. As a point of interest studio monitors (near-field, critical working speakers) typically use 4th order crossovers.
Note that single driver designs resolve all of these concerns.
