I'll say it up front. All things considered, I think active crossovers are better. This is a big topic and it's quite complex, so for now I will cover what I consider some of the stand out issues. First, I'll start with an opposing view.
Why passive crossovers might be better
Some argue that passive crossover components are in general more sonically transparent in mid and high frequencies and that active crossovers should only be used for bi-amping so that large inductors on bass drivers can be avoided. This is a point of view that can't be easily proven as in comparing active vs passive, there will always be many variables. I don't see any way that you could isolate "sonic transparency." So this will remain an opinion untested.
An obvious advantage to passive crossovers is that only one amplifier is required. This allows spending to focus on just one amp and this might be seen as a cost saving.
I'm trying to be objective here, but I see this as the extent of the "passive is better" argument.
Why active crossovers are probably better
Control and flexibility
One of the most significant advantages is that an active crossover has much greater power and flexibility. We can easily use steep slopes and a range of EQ and DSP options to gain precise control over the result. We can get tweeters to cross lower and we have far more power to get the most out of the drivers. Cone breakup is far easier to control. Dipole roll-off can be compensated for. Bass drivers can be matched to their box with precise control. All of this means clearly audible improvements and much less compromise.
Active systems have far better dynamics. A passive system will often see clipping caused by power hungry bass drivers, resulting in distortion and a harsh sound coming through the tweeters. An active system avoids this as the tweeter will have it's own amp. Clipping caused by the bass drivers will probably not be audible where with the passive system it causes the system to sound strained and harsh.
Each amp becomes a specialist and can be chosen for a narrower bandwidth. No need to compromise. You might combine a powerful pro amp for the bass, with your favourite amp in the midrange, and a low powered tube for the tweeter.
Many make the mistake of compromising their speakers simply to use a valve amp that they prefer. With an active system this is not necessary. Drivers can be chosen for ultimate performance in their bandwidth and efficiency becomes less critical. Many efficient drivers have a lot of coloration and problems - that's a high price to pay to use the amp that you prefer. A 7w valve amp will drive any tweeter to very high output levels.
This post is scratching the surface, but some of the best articles I've seen on active are here:
How to get started in active
First, have a look at your options. Starting out on a budget, I suggest MiniDSP which involves some light duty DIY. With a few more features and no DIY electronics, many choose Behringer Ultradrive. A step up in performance and a big step up in price is DEQX.
More about active options in detail >
You will need a basic measurement setup >
This will include a mic and mic preamp.
You will then need to obtain software and measure driver acoustic response and work apply filters until you can measure a nice flat response, but your journey won't end there.