Passive pickups have one big problem: they have a high source impedance. This means that they require a ridiculously high impedance input. A guitar premaplifier must ideally have an impedance of at least a million Ω (Ohms). Smaller impedances will cause high frequencies to suffer. As an extreme example, if you plug a passive guitar into a line-level audio input that has a 10 kΩ impedance, there is almost no high end.
Because active pickups incorporate an amplifier, their source impdance is very low. This means that the same tone profile will be produced by the instrument into any input: high or low impedance.
The high source impedance of passive pickups is also why guitar volume potentiometers ("pots") have high values like 500 kΩ. The value of a volume pot affects the tone: use a 250 kΩ, and there will be less high end. Active pickup systems can use small potentiometers like 25 kΩ or perhaps even smaller. These are physically smaller and more readily available in wider varieties.
The output of passive pickups depends on the strength of the magnet, number of windings and proximity of the string. A stronger magnet causes string pull, which can affect tone. The string being close to the pickup also affects tone. More windings reduce the high frequency response. Active pickups can provide lots of output without the artifacts of string pull or a rolled-off high end.
An attractive alternative to active pickups, however, is to install an on-board preamplifier in conjunction with passive pickups. A very effective preamp can be built using a single JFET transistor, powered by a 9V battery. A nice DIY electronics projects is to eliminate the battery by a phantom power system. The idea is that a DC voltage can be induced in the instrument cable, from which the guitar's on-board amplifier can draw current. I also seem to recall I saw a project once whereby someone built a guitar preamp entirely in the 1/4" plug of the instrument-end of the guitar cable, and had it phantom-powered. Every guitar you plug that cable into becomes active. Of course, that cable then only works with the special phantom power source. It looks like someone turned this into a product.