To understand why splits matter, you need to understand how defensive fronts align. Typically, most defenses are taught to align on the basis of where the offensive guys align, which makes sense because those defenders are trying to get through or around the blockers to get to the running back or quarterback. So defensive linemen and linebackers were told from the earliest days of football to align "on the inside eye of the guard," "heads up the center," or "on the outside eye of the tackle." The linebackers had similar instructions, though they aligned behind the offensive line. Over time, defenses got better at mixing up these alignments, even before the snap. We've all seen linemen shift from the outside eye of the guard to the gap between the guard and center, or simply align late. All this is designed to confuse blocking schemes.
So as offenses became more complex, it became necessary to give linemen rules that would allow the run play to be blocked no matter what games the defense played, and to do that you needed a nomenclature that could be communicated via playbook as well as on the sideline (or at the line) in the heat of a game. This system became known as the assignment of defensive "techniques" to each defensive player. The credit for it is typically given to Alabama's legendary coach Bear Bryant, though he gives much of the credit to Bum Phillips. Below is an example of the numbering system.
Note that this is not the same as "hole numbering," because it is about where the defender aligns not where the run is designed to go. Although it looks a bit confusing, this system is used at literally every level of football, from pee wee football to the NFL. Below is another diagram with slightly different nomenclature, though it also specifies the "gaps." (Hat tip to the USC Trojan Football Analysis site for the image.)
So now that we know that defenses align based on where the offense aligns, and we know that offenses identify defenses based on the alignment, we can discuss splits. It's a bit of an oversimplification, but the choice is basically between tight or wide splits. I begin with tight splits.
Tight splits are the most common. In fact, most people probably don't think of them as tight, but merely notice when they see "wide splits." Below is an example of a typical alignment.
The advantage of tight splits are easy to see: Linemen are close to each other so you can get good teamwork between them; there are few or at least narrow gaps between them; and the line is constricted to keep defenders away from outside runs and quick outside throws.
The teamwork part cannot be underemphasized. One reason that tight splits are so common is because zone running and slide protection is so popular today. Zone running requires linemen to step in a direction, double-team guys in their area, and then one of them works up to block the linebacker. If the linemen are too far apart, you cannot get a good double-team, and the play won't go. For slide protection, linemen slide into a gap, and work together to create a fence for the QB. Any unblocked rushers must come from the outside, as the priority is to prevent a blitzer or linemen up the middle.
The point about gaps is similar. But the point about constricting the line for outside plays is underemphasized. Most teams, when they want to run an outside option play or a sweep of some kind, will have their line condense in by cutting their splits. That way a fast runner can get outside quickly.
Wide splits are more interesting. Traditionally, the teams with the widest splits were option running teams. That might sound surprising, but the reason was is that they used a lot of man blocking rules (i.e. block your man, rather than zone an area). More importantly though, by splitting out, because the defense aligned on the basis of where the offensive linemen were, the guy the QB was reading was split out. So if on the triple option you wanted your QB to first read the defensive tackle ("T") and then the defensive end ("E"), you'd split your linemen out to give him more time to make each successive read. (Hat tip: Hugh Wyatt)
You also simply created wide running lanes inside by having your linemen split out so wide. If you watched the old Nebraska teams, while they didn't take enormous splits, they did have wide ones for both their inside option plays and inside man blocking runs.
But there's a new trend for wide splits, and that's with air-it-out passing teams like Texas Tech. Traditionally passing teams took very narrow splits to stop inside penetration, Texas Tech takes exceptionally large splits. Their rationale is a few-fold: (a) make the pass rushers come from farther away and enlarge the pocket, (b) open up throwing lanes for the quarterback, and (c) because they throw so much, all they need is a block or two to have an effective draw play -- the defensive ends aren't even really a factor. They can do this because they are almost exclusively a "man" pass protection team, just as the old Lavell Edwards BYU offenses were. (Indeed, Mike Leach's offense is a direct descendent from BYU's offense, he spent time there as an assistant, and many of his other coaches had experience at BYU as players or coaches when Edwards and Norm Chow were there.)
The obvious concerns are that if one guy gets beat in pass protection then there is no help, and also that there are wide gaps for linebackers to shoot through. For the latter, Tech feels like they can hurt that in other ways, through quick passes, screens, outside run plays, and traps. And they also feel that they can simply teach their linemen to be smart and reactive, and still stop that kind of penetration.
For the former problem though, the answer is simply that they have to have good blockers. They freely admit that they put their linemen one on one a great deal of the time, but their philosophy is that if someone gets to the quarterback, everyone knows who got beat. More and more teams have been adopting this strategy.
As a side note, I observe that Leach went to this trend after he got away from having a two-back formation as his primary one for passing downs. With a two-back offense you can stop a lot of overload passing threats to either side, but with a one-back formation -- Leach's current primary version -- the wide splits were necessary to take those extra rushers out of the play. For more on all this, see my old article here. And you can get a flavor for what Texas Tech does in the video below:
So in sum, the choice of what splits a team uses will vary by play. Some will rely on teamwork and overwhelming force to overpower the defense, others will play games with varying them to set up the play they have called, and others, like Texas Tech, build it into their philosophy. As a final thought, many of you might think: Hey, if you always go tight splits for outside runs and wide for inside runs, won't the defense catch on? The response is the same one Bill Walsh would give when he heard this concern: If you have built a tendency (like running inside whenever you go wide splits), you simply self-scout, figure that out, and then confuse the defense by breaking your own tendency. Some of his biggest plays came when he broke his own tendencies.