As others have said, sometimes 2 behaves differently from the odd primes and sometimes it doesn't. In my own work, the difference often comes from the fact that, working modulo 2, we always have x = x, which is not true working modulo an odd prime.
Here's a specific case where that makes a difference. I often work with systems that are anticommutative, meaning that there's a multiplication with yx = xy. Looking at the case where y = x, that means that x^{2} = x^{2}. That's fine and always true mod 2, but mod an odd prime it means that we have to have x^{2} = 0. The latter puts a significant restriction on what anticommutative structures are possible mod an odd prime that isn't present working mod 2.
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