I think 2 is not special, we just see the weirdness at 2 earlier than the weirdness at odd primes.

For example, consider Ext_{E(x)}(F_{p} , F_{p}) where E(x) denotes an exterior algebra over F_{p}. If p=2 this is a polynomial algebra on a class x_{1} in degree 1 and if p is odd this is an exterior algebra on a class x_{1} tensor a polynomial algebra on x_{2}. I say these are the same, generated by x_{1} and x_{2} in both cases and with a p-fold Massey product < x_{1},...,x_{1}>=x_{2}. The only difference is that a 2-fold Massey product is simply a product.

In what sense are the p-adic integers Z_{p} the same? One way to say it is that if you study the algebraic K-theory of Z_{p} you find that the first torsion is in degree 2p-3. If p=2 this is degree 1, and K_{1}(A) measures the units of A (for a reasonable ring A). If p is odd it measures something something more complicated. Another way to say it is that Z_{p} is the first Morava stabilizer algebra and there is something special about the n'th Morava stabilizer algebra at p if p-1 divides n. If you study something like topological modular forms, this means the primes 2 and 3 are special.

The dual Steenrod algebra is generated by \xi_{i} at p=2 and by \xi_{i} and \tau_{i} at odd primes. But really it is generated by \tau_{i} with a p-fold Massey product < \tau_{i},...,\tau_{i}>=\xi_{i+1} at all primes, after renaming the generators at p=2. (Again a 2-fold Massey product is just a product.)

I could go on, but maybe this is enough for now.