It appears to me that the problem here (wrt to both economics and climate models) is the assumption that we cannot predict anything if we cannot predict everything. That is all wrong. Although it is impossible to model any complex system in all its attributes with precision, that doesn't imply that we can't make any precise predictions whatsoever or that we're reduced to statistical guesses as Loki_1929 claimed. Even in complicated systems, there are some inferences which are non-obvious and which offer precise, accurate predictions, even if they do not predict the entire future state of the complex system.
For example, it may be impossible to predict weather because of its chaotic features, but this does not imply that there are no precise and regular features of climate. For example, the Earth will maintain an energy balance in the long run despite irregularities caused by weather. Although we cannot predict the weather, we can make some predictions about climate with near certainty.
Similarly with economics. It cannot be predicted when a recession will occur, or what the stock market will do tomorrow. But some things can be predicted with certainty. For example, it can be predicted with absolute precision that the purchase of T-Bonds by the Chinese government will cause the trade deficit to increase also and by the same amount, provided that other international debt purchases remain constant. Also, it can be predicted with certainty that a decrease in the bank reserve ratio will cause a proportional increase in the general price level. Also, it can be predicted with certainty that an increase in labor productivity will cause wages to rise and not unemployment. These predictions are highly accurate, and are not at all obvious (in fact the last one is actively disputed by many people).
The problem with both economics and predictions of future temperatures, is that theorists will attempt to make predictions even of things which they know can't be predicted with certainty. They figure that their guess is better than nothing. Thus economists will attempt to predict recessions, and weather forecasters will attempt to predict tomorrow's temperature. When the predictions fail, people unfamiliar with those disciplines will jump to the conclusion that "well the whole thing is just fucked then; it's too complex" and that we know nothing about either economies or future temperatures. But that is all wrong. Unfortunately those same people will subsequently ignore predictions which really are quite certain. At least with weather and climate, the disciplines are separated. With economics, predictions about recessions are grouped under the same label as (for example) trade theory.
Comparing economics to a real science is almost a mortal insult due to the lack of rigor in economics.
This is totally wrong. Economics is definitely a real science and is extremely rigorous. The problem is that economists have not successfully communicated which things they really know with certainty and which they're guessing about. With climate, climatologists have not successfully communicated the difference between weather (which cannot be predicted well) and climate (which can). In both cases, they have not successfully communicated that they do know some things with great precision even about very complex systems.