The relative size but also the structure and the process of fertilization. The sperm enters the egg, not the other way around. Mitochondria come from the egg, yes.
Also note that sperm vary in shape. Mammalian sperm have the familiar tadpole shape with a flagellum for swimming. Other species, such as nematodes, have ameboid sperm which move along surfaces rather than swimming freely.
> The sperm enters the egg, not the other way around.
I think this is wrong. My understanding is that the sperm cell membrane fuses with the egg cell membrane, forming a single container. The haploid nucleus of the sperm and the haploid nucleus of the egg then fuse. This processes is by no means fully symmetric (the egg has a tough coating on its cell membrane that must be dissolved by the sperm through the Acrosome reaction), but there is also no time when the sperm cell (delineated by its membrane) is inside the egg cell.
My point was not that there aren't many correlated asymmetries between eggs and sperm in particular parts of the phylogenetic tree (like mammals) that can be used for distinguishing them. My point was that there seems to be no fundamental asymmetry that applies to all two-sex sexually reproducing organisms; rather, there is a continuum of degree of gamete dimorphism that ends at organisms where there is no principled distinction between sperm and egg (isogamy).
I red this in some book I forgot the name of, so I can’t cite anything. But it was explained that female strips everything except for the DNA from sperm. This way there’s minimal risk of infecting the egg with the pathogens the sperm can carry. I’m not sure if it’s true just for humans or other organisms as well.
Also note that sperm vary in shape. Mammalian sperm have the familiar tadpole shape with a flagellum for swimming. Other species, such as nematodes, have ameboid sperm which move along surfaces rather than swimming freely.