This is a topic which can get a roomfull of sound engineers going, for sure. There are essentially 2 camps of thought which I have encountered, and they are diametrically opposed. One camp says that subs should be in-line to avoid imbalanced mixing and LF phase mismatches, while the other camp promotes bus-fed subs in order to keep the signal clean by giving the engineer the control to assign to the sub only those things truly belonging there, and this camp also claims the ability to handle phase mismatches. Let me state up front that the study and practical use of subs is a subject which needs far more than a short blog to adequately address. That said, I am going to attempt to simply compare here the pros and cons of in-line vs bus-fed subs as I see them.
Just so that it is said, so we are on the same page: the use of a sub speaker system with a set of tops, or tops and mids, is intended to make amplification more efficient. If different amplifiers and different loudspeakers are tasked with their own restricted set of frequencies, the result can be a clearer and potentially louder sound reinforcement system.
In-Line subs:
The in-line sub is connected from the main L/R outputs. Typically, the sub amp takes the L/R inputs and does a few things with them. They might be passed through to a set of L/R "through" jacks without any modification; they might be passed to a pair of L/R "processed" outputs, after applying a HPF; they will be summed together, processed though a LPF, and sent to the sub loudspeaker. Note that the F(c)'s of the HPF and LPF I refer to here will have some correlation to each other. At the simplest, these F(c)'s might be the same frequency, but in practice that is a simplification. In general, you connect the mixer L/R outputs to the sub L/R inputs, and then connect the processed sub's L/R outputs to the top speakers' inputs. Easy setup.
What you get then is that everything in the mix is divided between the subs and the tops according to the actions of the crossover in the sub's processing system. This assures that everything which has low frequency content gets the benefit of the sub. This includes any plosives and mic stand bumps which aren't adequately filtered by the mixer's low cut filter.
In a basic setup, if your subs and tops are essentially in the same plane (front to back), you generally assume that your phase is correct (but don't really assume this, measure it!). But if your sub (or subs) is in front of the stage, and your tops are elevated above the front edge of the stage, you already have 2 or 3 feet of difference between the plane of the tops vs that of the sub. This difference will begin to introduce phase problems, which need some time alignment to correct. Your sub amp might have a built-in delay to help with this time alignment, and it may not. The trouble is, depending on whether the sub or the tops is farther forward, your time alignment may need to delay the sub, or it may need to delay the tops. The point is, there is no inherant automatic phase alignment associated with using an in-line sub.
Bus-Fed Subs:
The bus-fed sub is effectively a separate speaker system from the tops (mids/tops), and as such LF signal is fed to this type of sub from a dedicated bus on the mixer (so the name, bus-fed sub). This type of connection requires not only the L/R cables going to the L/R tops or tops/mids, but also an additional cable (or more) from the mix console to the low frequency speaker system. Many consoles offer two main faders, one for L/R, the other for Mono/Sub. These generally do not operate as a connected pair, so must be adjusted manually while maintaining the relationshiop between the two. The sub again needs a LPF, either as part of the sub's amp or managed by or immediately after the mix console, and the tops or mids/tops also need a HPF coordinated with that LPF for efficient division of energy. As indicated a few lines back, this can be done by EQ circuits in the mix console, or the L/R and sub bus outputs from the console can be passed to an electronic crossover unit before reaching the amps. The setup is obviously more involved for bus-fed subs than for in-line subs.
In this type of system, you have the ability to select precisely which inputs at the console will be allowed to have any chance of reaching the sub. By assigning or un-assigning any given channel from the sub bus, you intentionally determine whether that signal gets the benefit of the sub. Those plosives and mic stand bumps I mentioned a few paragraphs back? If you do not assign any vocal mic channels to the sub bus, those undesirable sounds do not make themselves known in the low frequency power boost of the sub. Similarly, other channels which do contain LF energy, but it is undesirable to hear it, can be removed from the sub bus so as to not clutter that range with extraneous events.
There is an argument that with all those cables, the length of the signal path from the console to the tops vs to the subs is potentially different, and it is argued that this difference reults in phase mismatch. I suggest, if the fraction of a microsecond introduced by mismatched cable lengths is truly of any concern, what is the difference between that state and that of daisy-chaining the tops from the sub via an additional set of cables? The signal path length to the subs is different than it is for the tops. And again, where the speaker cabinets are located can introduce time alignment issues which in turn requires a delay for one or the other - again, how is this different from the in-line method?
The result of bus-fed subs is an arguably cleaner LF signal allowing greater amplification with less distortion, while requiring a more complex setup than its in-line counterpart.
Which system is better?
If one was impirically "better" than the other, we'd likely not have a choice in how to set it up. But we do have a choice, so you as an engineer need to answer that one WRT your needs, situation, and philosophy. For me, I've tried both, but I prefer bus-fed primarly because the overall mix sounds more defined to me, but I have used in-line on occasion when certain criteria have made it necessary or at least expedient.
Peace and Good Mixes....