The obligate aerobes need oxygen for their survival, while the obligate anaerobes do not. The obligate aerobes are the species that attain the energy for the process of aerobic respiration with the help of oxygen as the ultimate electron acceptor for the electron transport chain.
On the other hand, obligate anaerobes are the species, which get poisoned by the usual levels of atmospheric oxygen, and thus, get killed in the existence of oxygen.