The "O" is pronounced the same way--as a long "O"--in both words. (Shrug) I'm a U.S. native speaker and I DO pronounce the "th" in "clothes." I pronounce "clothes" by saying "clothe" and adding an "s" sound on the end. "Th" is one consonant, "s" is a second, and whenever I pronounce two consecutive consonants there's just a trace of something in between, a "schwa," I guess. I consider the pronunciation "close" to be slightly sloppy diction, but so common as not to be an error. I see that https://ahdictionary.com/word/search.html?q=clothes&submit.x=58&submit.y=14 actually lists BOTH pronunciations, "klōz, klōthz." Yes, it is difficult to pronounce that th+s combination. As evidence that "klōz" is not always considered standard English, I would point out that it is sometimes written as "clo's" in spoken dialog. Thus, Mark Twain: "I know you took it off, and know it by a better way than your wool-gethering memory, too, because it was on the clo's-line yesterday—I see it there myself."

"Clothes" is pronounced exactly like "close" without the "th" making a sound. "Clothe" is different. You do hear the "th" sound in "clothe", but it's not the typical "th" sound you hear in "the" or "that" It's softer. "Th" acts as one sound written as two letters, so any rule about adjacent consonants doesn't apply.