In Greek Mythology, their are a few special heroes (Diomedes being one) who, after Death claims them, have a shrine/temple build in their honor and for worship. Was the worship they received similar/same to that of a god, or completely different? In what ways?
In Proclusean hierarchy heroes were higher ranking than mortals, purified souls, daimons, lower than angels, archangels and Deities. They received worship and commemoration alike to the Gods. In fact they were - for example - Herculean mysteries and Herculean temples, stemming from the Eleusis rites (Heracles was a succesful initiate of these mystery schools). It is difficult to tell from a qualitative perspective what was the cognitive and spiritual way of expressing the worship, but I guess that in these days, when mixing and exchanging Deities was common, one might even have such instances when a hero received greater worship than a higher Deity in a different region.