First the similarities: a) both exist and are recurrent; b) there are local/global as well as long/short cycles; c) both coexist with a high frequency of outliers (whether bumper profits or extreme temperatures).
Second the differences: a) economic cycles are mostly man-made, climate cycles are mostly nature-made; b) economic cycles can be significantly smoothed (through Keynesian policies), no equivalent climatologist has shown that man can smooth climate cycles; and c) the side effects of tempering with economic cycles are known, but not so for climate change.
Third scientific attitudes and effort: a) for more than a century economists have dedicated a large effort to study business cycles, the same with climatologists but given the research costs of the later their effort is probably lacking; b) neither economists nor climatologist have an undisputed body of consolidated knowledge; and c) the two sciences are still dominated by fads and ideology.
Fourth the involvement of politicians: a) electoral politics has always relied on economic cycles (it’s the economy stupid…), but only recently turned to climate change; b) its involvement in economics cycles is justified since these are man-made, but is questionable for nature-made cycles; and c) politicians should decide whether it is worthwhile smoothing the business cycle, but should delay acting on climate until its science knows enough about the risks of changing nature.
Finally, where do I stand on climate change: a) climate cycles do exist; b) there is little we can do to smooth them; and c) climate change should be studied scientifically, not used to bring back old mistakes based on ideas of self-sufficiency, protectionism, and huge tax-payers handouts to sponging self-serving interested groups.