You need to look at the spectrum. There you will see lots of differences, though they're both somewhat similar to black body radiation. Just Google solar spectrum images and you'll see it. The spectrum of fire will depend on what's burning and at what temperature it's at. Basically different things are burning so their color will be slightly different. Sunlight peaks in the green of course because it is very very hot - much hotter than your typical campfire, which will have a spectrum shifted more towards the yellow and red with less in the blue. Additionally sunlight will have several absorption bands for things in the atmosphere like Co2 and water.
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