In guns of equal strength, it's basically a case of six in one hand, half a dozen in the other as they'll both do the same thing.
Often it is quoted that .45 Long Colt, when loaded to its full potential in an appropriate gun, will outperform the .44 Magnum. Specialty rounds from boutique makers like Buffalo Bore and Grizzly are certainly able to outperform the run-of-the-mill .44 Magnum loading, but what's forgotten is that the vast majority of .44 Magnum ammunition is made to be safely used in any so-chambered gun. If we look at boutique ammunition loaded specifically for the biggest and strongest .44 Magnum revolvers such as is available from Buffalo Bore, Grizzly, and Garrett, we find that the difference in ballistics between the two cartridges is so small that nothing you could ever shoot would be likely to notice the difference.
Also, we see the difference in diameter and frontal area often quoted, but no one ever seems to stop and think about what, exactly, that diameter gets you. While it is true that, for the same OAL, a .45 Long Colt can use a slightly heavier bullet, it is often overlooked that the smaller diameter of the .44 Magnum allows it to penetrate just as well, if not better, with a lighter bullet due to its better cross-sectional density. As for the difference in frontal area, lets be honest here: the true top-end loadings in both cartridges are usually loaded with tough non-expanding bullets. This is because, on the type of game that these loadings are intended for, the only two factors that really matter are placement and penetration and no amount of frontal area can make up for shortcomings in those two areas.
Finally, the issue of pressure is often raised. While it is true that .45 Long Colt can attain its performance at lower pressure than .44 Magnum, I've yet to see anyone explain why exactly it makes a difference. I've never seen anything shot that asked what the pressure of the cartridge was and, since the ballistics of the two cartridges can be so close, I don't think it matters so long as the gun can handle said pressure. The other thing to consider is that while .45 Long Colt may be a lower pressure cartridge, comparable .44 Magnum revolvers are able to safely handle higher pressures due to thicker chamber walls. That very issue is the reason that Elmer Kieth abandoned his experiments with .45 Long Colt and turned his focus to .44 Special: at the time there were no .45 Long Colt revolvers that could safely handle the pressures Elmer wanted to work at.
Which cartridge is better is more dependant, I think, upon the user and his application. While a handloader can produce .45LC ammo just as easily as .44 Mag, someone who does not handload will be able to acquire .44 Mag ammo easier and cheaper. Also, while boutique ammo makes the ballistics very close, most "off the shelf" .45LC ammo will deliver ballistics much closer to that of .44 Spl or .45 ACP than .44 Mag.
Also, I'd take .44 Magnum over .45 Long Colt for self-defense use in a DA revolver due to the differences in the diameter of the rim. .45 LC has a very small rim, almost more like that of a semi-rimmed cartridge such as .32 Auto or .38 Super, because the rim was designed originally only for headspacing in a gate-loading revolver. .44 Magnum, on the other hand, was designed from the ground up to be used in revolvers with simultaneous extraction. Thus, .44 Magnum has a more prominent rim that gives the extractor more surface to bear against. The reason that I view this as important is because the small rim of the .45 LC is more likely to "jump" the extractor of a DA revolver and tie up the gun until it can be removed (a task that is neither quick nor easy). While this may be just a minor annoyance on the range, it could create a very bad situation if you're trying to reload your revolver while someone, or something, is trying to kill you. Of course, the diameter of the rim is not an issue for someone using their revolver for a non-critical application like hunting nor would it be an issue for someone using a gate-loading SA revolver.
. Beyond that your into 'use your rifle' range.