The first way of differentiating this is by the clinical examination - for example, nystagmus (jerky eye movements) are usually present in both but nystagmus that does not fatigue, is vertical or rotational, not associated with severe nausea etc is more likely to be central, and the opposite for peripheral.
Also, a detailed analysis of eye movements with and without video-nystagmography will aid the oto-neurologist in making the diagnosis and bring out the eye movements necessary to make the diagnosis. This is better than ENG or electronystagmography, although ENG may provide additional help.
Dilantin even at therapeutic levels can cause those symptoms in individuals suseptiple to them - we go by clinical symptoms/side effects rather than levels/doses a lot of the time. Dilantin is associated with cerebellar atrophy in long term use, also osteoporosis, so then newer antiepileptic drugs are preferred in terms of side effects.
CP angle tumors can cause dizziness by compression of the vestibular nerve. Residual symptoms depend on the degree of damage done before the compression was releived. Recovery if symptoms are present for over 12-18 months after the compression is releived would be unusual.
Aspartame has been associated in studies with seizures and dizziness. An association is not quite proof, but may be a useful thing to avoid in patients who already have seizures or dizziness.
MRI of the inner ear is generally not done by neurologists, I cannot speak for ENT.