Well, you’re going to get a lot of different answers. Some of them are just going to be flat wrong (“relievers have to have control” compared to starters is one), but there are a lot of arguments to be made.
The basic idea is that the best pitchers should be your starters, since over the course of a season a starter is going to get many more innings than any reliever will. So any very good pitcher is going to at least be considered as a starter; if he can keep up the same level of effectiveness or close to it, he’s much more valuable throwing 220 innings than 75.
It doesn’t always work out, though, because not everybody is as effective over the long haul as they can be in short stints. Sometimes that’s because they really only have one or two exceptional pitches. A one or two pitch pitcher, say a guy who can throw a good fastball and a great slider, but whose other pitches are substandard, is much more likely to get whacked around if he pitches six innings than if he pitches only one or two. Batters can adjust, and if you don’t have any way to mix them up and disrupt their timing, you’re either going to throw way too many pitches trying to blow them away, and tire out and walk batters, or you’re going to get hit because you aren’t fooling them.
Bullpen vs. starting rotation a risk-reward thing that plays out pretty frequently with young and very talented pitchers. Neftali Feliz of the Rangers throws 97 MPH, but he basically throws fastballs all the time, and essentially has no third pitch. Good enough to wipe people out for an inning at a time, which makes him a great closer right now. But with a couple years of seasoning, could he be a great starting pitcher who still throws 97 MPH?
The game you watched last night had a similar guy. Daniel Bard pitched the 9th inning for the Red Sox. He’s 26 and has been a great relief pitcher for the last few years. He’s a big dude and throws 97 MPH. But he throws fastballs and sliders 95% of the time, and that means he’s basically a one-inning pitcher, like you saw last night - 27 pitches in one inning is a lot of work, and you can’t do that if you’re expected to throw 8 innings like Beckett, who also can throw that hard. So, leave Bard in the bullpen where he’s been dominant, or see what he can do as a starter and risk him being ineffective? People will argue both ways. Some of the all-time great relief pitchers have been former starters whose skills were better suited to relief, too, so arguably not everybody should be a starter even if they can do it. We don’t know what Mariano Rivera would have been as a starter, which is how he started, but we do know he’s the best closer of all time.