Females are often bigger and heavier than males (not in our species) as they have to invest more resources in producing offspring – the males contribute just sperm, which is cheap. That’s the case for a lot of mammals, some birds of prey, &c.
The sexual arms races for extraordinary characters have already been mentioned.
Other advantages of sexual dimorphism can be seen in extreme cases:
* some deep-sea fishes have big females but very tiny males living parasitically on them. The deep oceans are dark and vast and once you’ve found a mate you stick to it. Literally;
* bees and ants have practically created three “sexes” (or castes): a big solitary queen busy laying eggs, sterile workers (actually females) foraging for the queen and the larvae, and temporary males which are just vectors for the queen’s genes. A kind of natural taylorism, efficiency by specialization. 😉
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