Soft Proofing Doesn't Necessarily Mean Correct Color
"Soft proofing" refers to the ability of software to show you what your image will look like printed, before you actually print it. So if the colors in your prints still come out wrong, what's the reason?
To begin with, we need to make sure we're all on the same page here. Soft proofing is a technique built on the foundation of color management. So it's with that foundation that we need to start.
No two devices capable of reproducing color do so in a quite the same way. In a perfect world, perhaps they would or should, but we don't live in a perfect world. Every device has its quirks. No two paper stocks have the same brightness and reflectivity, nor do they absorb ink in quite the same way or to the same degree. And ink manufacturers haven't yet developed products that produce uniform color gradients across the spectrum proportional to the amount of ink applied. Companies that make ink expend a great deal of effort on expanding the range (gamut) of colors they can produce, even when they have to thereby sacrifice some degree of linearity in color rendering. And then there's print longevity to optimize for as well. An ink that renders uniformly beautiful color but that fades quickly won't sell well. Let's just say that printer, paper and ink companies have a lot of variables to balance and there are always tradeoffs. Indeed, calling the color rendering particularities "quirks" is perhaps being too kind.
But if those color reproduction quirks can be described adequately, they can be compensated for via software. Color management is that description and compensation mechanism. The full details of how it all works is beyond the scope of this article, but suffice it to say that color management is an industry wide standard that allows different devices and software applications to interact properly so long as everybody involved follows that standard.
A color profile is the component within the color management standard that serves as the description of how a given device reproduces color. It describes those "quirks" so that they can then be compensated for. If the driver for your printer knows ahead of time that it needs to compensate in a particular way in a particular situation via the profile, it can do so automatically so that you see only corrected color, not what would have been printed without correction. It's a wonderful thing.
But since color management is a standard, that same printer profile can also be leveraged in other ways. For the topic at hand today, that same profile can be used directly by Lightroom or Photoshop to simulate what your printer driver would do or will do when you print, but without consuming any paper or ink. You see the simulated print directly on your monitor display.
And this brings me to the most common reason why soft proofing doesn't always work the way you expect: your monitor. You may have an absolutely incredible printer capable of producing stunning prints worthy of hanging in any gallery or museum, but all that technology will be severely crippled if you don't pay attention to your monitor. I mean, how can your monitor display what your print will come out looking like if it can't render color properly itself? It too has its quirks.
Modern LCD monitors have come a long way in their ability to render color right out of the box. They look gorgeous and your images will likely look gorgeous on one. But that doesn't mean they will look correct. That doesn't mean they will look the way they actually do. Unless you compensate for the quirks of both your printer and your monitor, soft proofing can't possibly work as it should.
But you say your images look fine on your monitor but still print wrong. That may well be so, but more than likely you have used that quirky monitor to edit your images, not just to soft proof them. And if this is true, you have probably messed up the color in your images without even realizing you are doing so. As you think you are optimizing your images and getting them ready to print, you are actually distorting their color because you are judging what looks good by means of that quirky monitor. Of course once you do, your images will look fine, but they're not really. Rather than compensating for the quirks in how your monitor by using a monitor profile, you will have instead embedded that correction into your images themselves. If your monitor makes things look too green, you can still get your images to look OK by pushing their color a bit towards the magenta side. A slightly magenta image viewed on a monitor that makes things look a bit too green will appear OK when it's not actually so. What's wrong with one will appear to correct for what's wrong with the other.
But when you ultimately send that magenta image to your amazingly good printer it will of course come out with that magenta cast still intact because your monitor will no longer be intervening to compensate. Your images will only look good on your quirky monitor, not on your printer. And when you someday buy a new monitor, all your images will look wrong all over again. Its quirks won't be the same as those of your old monitor. Once you fiddle a bit with the controls on that new monitor, you can get it making things look too green just like your old one, and thus your images will look OK again, but you'll still have trouble printing since neither your old nor new monitor will be there influencing the result.
There's really no substitute. You must profile your monitor to put an end to this problem. There's really no other way. You need to know that your images actually look the way the way you think they do. Only once you are able to compensate for your monitor's quirks can it properly utilize soft proofing to show you what your prints will look.