So, according to Other blood colors it is entirely possible for other blood colours to evolve based on what base they use, but why? What would be the benefits of these different bases? What purpose would they serve? What conditions would find it beneficial to evolve these?
Why we have red blood
From the data given in the answer to that question, Haemoglobin is by far the most efficient carrier of oxygen that we know of. Therefore, if the efficient oxygenation of blood is important to the species, then there would be no advantage (Well, very little, see below) to having different colour blood. At best it would be a disadvantage as you would find exercise a little bit more difficult and you'd have to breath faster to get enough oxygen into your body. At worst it would be crippling because the smallest increase in the body's oxygen requirement would be impossible to keep up with.
Why other species don't
Look at the example species. Mollusks, for instance, while they use oxygen, don't have heavy usage, so the evolution of a highly efficient carrier wasn't necessary. Therefore developing a dependency on consistent iron intake would be a disadvantage - no reason for haemoglobin. I've tried searching to see whether there is any documented anaemia equivalent for Hemocyanin, but surprisingly noone has researched cases of anaemia in mollusks, how remiss of them. Presumably a hemocyanin-using life form would still have to ingest copper in order to create the oxygen carriers, but this could be easier for mollusks than iron.
Why one might not have red blood
There is one major consequence of our red blood that could be considered a disadvantage in some conditions
Using haemoglobin as your oxygen carrier requires that you ingest sufficient iron to keep the concentration in your blood up - failure to do so results in a dramatic reduction in the oxygen-carrying ability of your blood, otherwise known as Anaemia. So the simplest reason to have different colour blood is if ingesting iron isn't something you find easy, and/or oxygenation isn't as important to you as a species.
For humans (or another life form of similar structure with a significant oxygen requirement), you could theorise an environment in which iron is impossible to ingest, either due to other physiological changes (toxic for some reason), or environmental absence (no iron to ingest). I found an article Here that implied that Hemocyanin has been used or at least suggested as a potential cure/helper for pregnant women suffering from anaemia (iron deficiency is very common during pregnancy as it can completely screw with your ability to keep food down and due to the extra nutrients required by the growing foetus). The extra oxygen carrier in the blood supplemented the low hemoglobin supply enough to prevent anaemia.
In theory, we could use hemocyanin in our bloodstreams if for some reason we couldn't produce hemoglobin. The oxygenation would only be 40% as effective, but there are plausible situations in which this might be ok. A planet with no iron and lower gravity for instance could theoretically work - less muscle mass means less oxygen required.
You could even have a doomsday human situation where iron is not available for some reason and people are having to use hemocyanin to survive. They'd probably still be pretty damn sick (See here for some details on hemoglobin levels, having an equivalent 40% level would make you seriously anaemic), but you might be able to counter that by flooding your system with hemocyanin, which would certainly give you blue blood. However, there would be other interesting side effects to consider, like Copper Toxicity/Metal Poisoning and Oxidative Stress which would potentially kill them anyway!
You can imagine a feudal society that would form - powerful people hoarding the last of the iron to keep themselves healthy, middle class getting by with treatments to help with the poisoning from the hemocyanin, peasant class basically crippled. The Blue-bloods would inevitably become smaller in stature and less powerful as their bodies wouldn't be able to afford the oxygen for lots of muscle, causing a physiological rift. Maybe, after an awful long time (or with a little scientific help), people from the hemocyanin side would naturally evolve an internal production and copper affinity. No longer dragged down by their iron deficiency, that might change the balance of power...
I can think of several reasons.
As @IStanley mentioned if iron wasn't available in sufficient quantities in the organism's environment, then another oxygen transport molecule using other metal ions would be required.
The different oxygen transport molecules operate at different efficiencies under different environmental conditions.
Low oxygen partial pressures
On the chart, it looks like Iridium (Ir - a platinum like noble metal) based oxygen transport molecules work better than hemoglobin at very low partial pressures of oxygen. So in a nearly anaerobic environment, it would work much better than hemoglobin would.
Different temperature regimes
The chart above only shows the variability in efficiency for some of the oxygen transport molecules. Presumably they all vary in effectiveness over temperature as well as oxygen partial pressure. It is therefore safe to assume that, like oxygen partial pressure, different oxygen transport molecules will become more effective at lower or higher temperatures than human body temperature (e.g. 37 C / 98.6 F).
All organisms will be restricted to using only those elements easily available in its environment. Although iron is universally about 100x more abundant than cobalt and many more times abundant than some of these other elements, you could easily envision environments in which iron is locally unavailable or less available than these other metal ions.
Other environmental factors will also influence the function of those molecules. It's hard to tell based upon this single chart but it looks like the following observations may be true.
At low oxygen partial pressures (nearly anaerobic environments), Cobaltodihistidine may provide better oxygen transport than Hemoglobin.
At high oxygen partial pressures, Iridium based molecules may work better than Hemoglobin.
At low temperatures, Coboglobin may work better than Hemoglobin.
At high temperatures, presumably some other oxygen transport molecule becomes more effective.
Some of the references from this answer about the colors of blood provide interesting reading material on the subject.