Do 'autopsy' and 'post-mortem' have essentially the same meaning?

For the most part, yes, "autopsy" and "postmortem" are used interchangeably in American English. However, there is a linguistic difference. "Autopsy" is a postmortem examination of a human body, generally to determine the cause of death. "Postmortem" simply means "after death" and can be used in a variety of instances where human bodies are not being examined. For instance, a veterinarian can perform a postmortem exam on a dog or cat; this is generally referred to as a necropsy.

Also, in the slaughter industry, the examination of carcasses after the animal is humanely killed, skinned and eviscerated is called the postmortem examination or postmortem disposition.