Magnitude mainly refers to abstract, uncountable, measurable, intangible, immaterial/nonphysical nouns, thoughts/ideas, energy, motion, actions (time), conducts, behaviours, function, importance, significance, greatness, brightness (mainly of stars), grade, degree, power, force, capability, measure, gravity, seriousness, toughness, worth, value, volume etc (adj. a little/some, a lot/much, important, significant, great, bright, powerful, forceful, capable, measur-ed-able, grave, serious, tough, worth(y), valu-ed-able, etc). “We did not realize the magnitude of the problem.” "A problem of the first magnitude". “The quake reached a magnitude of 7.1 on the Richter scale.” "They tried to predict the magnitude of the explosion"; "About the magnitude of a small pea", “The star varies in brightness by about three magnitudes”, “ A star of the second / third magnitude”
From Latin: Magnitudo: magnitude, greatness, importance, vastness, extent, bulk, Intensity.
Size mainly refers to concrete, countable, numerable, tangible, material nouns, physical objects with extent, body, volume (3D, of three dimensions), etc, (adj., a few/some, a lot/many, little, big, small, large, low, tall, short, high, long etc) “The facilities are excellent for a town that size.” “The kitchen is a good size.” "A wolf is about the size of a large dog" "He wears a size 13 shoe"
See, however: "That's the size of the situation"; "She hates me, that's about the size of it"