The purpose of this piece is not to focus so much on anything fantasy-related, although if anything I hit on here helps you with line-up decisions, well then that's just gravy. But what I really hope to do here is to highlight these two surgical procedures for our baseball players who read the site and give you a little more background as to what they are specifically and do a little compare/contrast. All of this from an athletic trainer's perspective (and fantasy player as well) to hopefully give you more insight when you hear one of these terms or a teammate suffers one of these injuries in the near future. Because it WILL happen eventually. Fair warning: there will be some medical jargon tossed around, but hopefully not too much that it gets boring. 

So here we go......

First, the labrum. There is actually a labrum associated with the hip joint. Both the hip and the shoulder are ball-and-socket joints, with the end of one bone forming a "ball" that fits into another bone's "socket." The labrum is a piece of cartilage that serves to deepen the socket. The hip is a much deeper socket than the shoulder, which allows for more range of motion (i.e. it allows the shoulder to throw a ball overhead). For the rest of this article, the term labrum and labral repair will refer to the shoulder labrum.
The shoulder wasn't made to throw a ball in an overhead motion with the force and frequency that a baseball pitcher's position requires. It can do it, and do it very well, but the odds of something breaking down over time or of something "abnormal" developing are very high. When this happens, sometimes a surgical procedure is required to repair or "clean-up" the damage.  
A typical injury to the labrum is a SLAP lesion. SLAP: "Superior Labral tear, Anterior-to-Posterior" refers to the direction of the cartilage tear. Symptoms for a pitcher include loss of velocity, a dull ache, location issues, and trouble sleeping. Often there is no "I felt it on one pitch" complaint; it's more of a build-up over time.
The picture above shows the tear before and then after the surgery to repair it. Once it's torn, it's torn—there is no "letting it heal on its own" thing here. It must be repaired, either arthroscopically or through an open procedure. Does it work? Yes and no. 
Almost 1/3 of the pitchers in baseball who have this surgery still report pain after the rehab process. Many experts regard a pitcher with a labral repair as a high risk, and sometimes write them off. Not as many pitchers with this injury come back to full capacity like Tommy John patients do. Let's do a quick summary of the Tommy John surgery and then we can get back to the discussion.
Tommy John Surgery (TJS):  We've all heard this described in detail—basically a pitcher's elbow gets to the point where it cannot absorb the force required to throw a baseball, and a key ligament tears. The ligament in question here is the ulnar collateral ligament (UCL). It's the ligament on the inside of your elbow right near the bone sticking out (go ahead and touch your elbow to see what I am talking about). Over time, the ligament wears down a little, but sometimes there is a "one-time pop or pain" after a throw, signifying to the observers that an injury just occurred.
Misconception: Pitchers think they come back even better than they were before surgery. Well, this is true, but only in that they are better than they were as an injured pitcher. Thanks to certain media articles and some success stories, there is a thought out there that pitchers come back with even more of performance edge—they throw harder, more movement, etc. This is simply not true. What happens is that they now have a healthy arm and can throw the ball again without pain.
While TJS pitchers come back at a high success rate, and labral repair patients have a lower rate of success, it is worth noting that "success" needs to be defined. The goal of TJS is simply to repair the elbow to allow for the patient to return to pitching a baseball...NOT to allow the pitcher to resume putting up quality numbers. Similarly, a labral repair in the shoulder is designed to decrease pain, and allow the pitcher to throw a baseball again at the big league level. An interesting piece about a study actually shows that TJS pitchers do not resume their former statistical performance levels prior to surgery. While the study has limitations, I think it's a great statement countering the notion that pitchers are "better" after TJS.
But why is it that, on the whole, TJS pitchers come back at a higher rate than when compared to the labral repair pitchers? Great question.
I think a lot has to do with the simple fact that the shoulder is such a complex joint, especially when compared to the elbow. When you throw so many factors together in one joint (more tissue, more range of motion, more overall joint interaction, etc.) and try to repair one thing, no matter how good a surgeon is, there is just more risk. I would also have to say that we have a lot more data on TJS (first one was done in 1974). Time will reveal more research and better surgical methods for the shoulder, hopefully increasing a pitcher's chances for a full recovery. A promising research project on labral repair, funded with a grant from MLB, could reveal some helpful info. 
Comparison chart comparing labral repair vs. TJS.
Also, no matter how good the surgeon is, the rehab & therapy process is just as important. Mess this part up, and nothing will work great. Compliance with a rehab protocol, both on the part of the AT/PT team and the patient, is critical to the success rate. I personally think it's easier for a patient to overdo some things with the shoulder joint in the healing process than it is with the elbow, as I have seen this in my own rehab experience with pitchers. MLB teams have outstanding rehab staffs and equipment, but that doesn't always guarantee that the pitcher is doing what he is supposed to do.