Are an EEG test and a sleep study test different?

Yes. Eeg records brain activity alone (save for one lead to record the heart), while sleep studies also record leg movement, breathing, oxygen levels.
Yes. An eeg test is to look for evidence of seizures. A sleep study test is to look for sleep disorders. A limited eeg is obtained during a sleep study in order to determine the stages of sleep the person is in, but the eeg is not as complete as one done to look for seizures. Other measures during a sleep study include breathing and muscle activity, which are not measured during a standard eeg.