S1. Both do general equilibrium analysis, trying to understand the interaction between parts of a whole system, with lots of positive and/or negative feedback loops.
S2. Both have got time-series data that show how different variables are correlated. We try to explain those correlations by building models. Sometimes we can use evidence from elsewhere to help us pin down the parameters in those models.
S3. Both rely on natural experiments to generate that data.
S4. Both are politicised. "Wrong" policies can cause great damage; and people's beliefs on macroeconomics/climate science tend to correlate (imperfectly) with their political beliefs.
Those are the similarities I can think of.
What about the differences?
D1. Macroeconomists are trying to explain people; climate scientists aren't. People are harder to explain. In particular, people's behaviour depends on what they expect to happen in future, so we have leads as well as lags in our causal modeling. That makes our job harder.
D2. I think we have more useful data. If we want to look at the effect of money on inflation, we can look at different countries, as well as different historical episodes. If they want to look at the effect of CO2 on temperature, they only have world data. And we have lots of natural experiments in historical memory. That makes our job easier.
D3. Some of our data is generated by policy-makers who deliberately try to minimise the signal/noise ratio, such as when central banks control monetary policy to try to keep inflation on target. That makes the data useless as a natural experiment. On the other hand, if central banks succeed in keeping inflation close to target, that suggests that whatever macroeconomic theory the central bank is using is probably approximately right.
D4. Macroeconomics is older than climate science. The theory that money causes prices is older than the theory that CO2 causes temperatures. The popularity of the quantity theory has waxed and waned and waxed and waned over the years.
D5. To talk about "mainstream" or "consensus" macroeconomics is not totally meaningless, but there is also considerable disagreement between competing approaches. If we define it loosely enough, some version of the quantity theory of money for example would probably be acceptable to most macroeconomists; but a significant minority would reject it. It's hard to think of any non-trivial part of macroeconomics on which there is near-universal near-certainty.
Those are the differences I can think of.
I've spent the last few days reading about the release (by hacker or whistleblower) of emails and files from the University of East Anglia's Climate Research Unit, and can't help but try to think of the macroeconomic equivalent. I don't think of macroeconomists as perfect models of scientific rectitude. But most of our fights take place within the academy, and are mirrored by fights outside the academy. There's almost as much variance inside the academy as there is between inside and outside the academy. That's the main difference between macroeconomics and climate science.