Why does pistachio ice cream not taste like pistachio nuts?

Firstly, let me disagree and say that I have tasted some (amazing) pistachio ice creams that tasted very much like the actual nuts. That said, I can think of several reasons why the ice cream might taste differently from the nuts:

• Artificial flavours, like you suggest. You might be able to figure out from the ingredients list if these are used.
• Salt. If you have only eaten salted pistachio nuts, you might have a different benchmark for pistachio flavour. Salt is a flavour enhancer, after all.
• Temperature. The temperature of food and drinks radically changes how it is perceived when tasting. This is why (bad) coffee is OK when hot, but terrible when at room temperature, and why melted ice cream tastes much sweeter than the frozen stuff.

A final side note, which I include mostly because I find it interesting: I used to think 'banana flavour' was based on a different type of banana than commonly found in stores. Doing some searching for this question, it turns out I was wrong.

This adds a couple more slightly speculative reasons to L.Schoon's list, which I commend.

Solubility: In ice cream, the pistachios are blended in with water and fats. When you eat them whole they're not. Any fat-soluble flavour compounds will be much more available in the ice cream than when chewed (and mixed with saliva, i.e. water). Some flavour compounds are definitely much more fat-soluble than water-soluble, such as capsaicin. Actual sugar and salt are of course very soluble in water.

Mouthfeel: this has a complex interaction with taste, but is all part of the experience of eating - think of the difference between buttered toast, dry toast, toast with jam (with/without butter).

These actually overlap in practice, and there's a whole field of scientific literature on the subject; one relevant paper is Effect of Fat Level on the Perception of Five Flavor Chemicals in Ice Cream With or Without Fat Mimetics by Using a Descriptive Test. The main conclusion for this question is that some flavour compounds were detected more strongly in full-fat ice cream, while others were detected more strongly in reduced fat recipes.

Most pistachio ice creams are made with some, if not most or even all, almond and almond flavoring in addition to pistachio flavoring. That is largely due to the cost difference in the two products, as well as the difficulty in processing pistachios, and probably also the fact that almond flavoring is common and generally well tolerated (by people not allergic to it, anyway!). The almond may be hiding behind "natural flavors", as the most common almond flavor (Bitter Almond, or Benzaldehyde) (which also doesn't taste like the almonds you might eat, which are sweet almonds).

For example, even Ben and Jerry's Pistachio Pistachio ice cream contains some almond flavoring!