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It's not really a straightforward win for either one - it depends on how you use them. Linen is the clear winner for dry heat because it breathes like almost nothing else and air passes right through it, but it wrinkles like crazy and sometimes can feel a bit stiff when starched. Cotton is more versatile - it holds moisture better than linen, so if you sweat a lot or the air is really humid, linen might not absorb enough and you'll stay wet against your skin.
The thing is, a lot of people don't realize this, but the real trick is the weave: a lightweight, soft linen beats heavy cotton, but an oxford-style cotton or even a linen-cotton blend is often the best compromise. If you're buying linen, get the kind that feels like a sheet of paper rather than the rough stuff, and if you want cotton, look for long-staple fibers (like pima or supima) that are cooler than standard fabric. A loose, open weave is what you're looking for in both cases - the less dense the fabric, the cooler you'll actually feel.
Linen is cooler than cotton because it absorbs moisture faster and releases it just as quickly, leaving you dry. Cotton holds more moisture against your skin, so it tends to feel hotter. If you're in the sun all day, linen is the better choice.
But cotton has its advantages: it's softer, less fragile, and wrinkles less. If you can't stand clothes that crease easily, cotton is more practical. Ideally you'd wear both in summer - linen when you know it'll be hot and humid, cotton the rest of the time. Really, a lot of summer clothes are a blend of the two anyway, which gets you the best of both.
Both have their pros and cons, but if I had to choose based on how you'll be wearing it: linen wins if you're out in the sun all day, cotton if you're spending hours in air-conditioned spaces (linen wrinkles like crazy and looks bad in an office). One trick nobody mentions is mixing the two: a lightweight cotton tee under an open linen shirt gives you linen's breathability without the heat that pure cotton traps against your skin when it gets soaked with sweat.
When you're outside in high heat, linen is objectively better - I notice it when I go to the mountains in summer and after hours in the sun, linen doesn't stick to me the way cotton would. But people always forget that linen wrinkles like crazy and if you're the type who cares about looking put-together it's a hassle, while cotton still looks decent even when it's creased. If you can tolerate the wrinkles, linen wins hands down.
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