How can I tell if my grandmother has alzheimer's disease or vascular dementia?

Doctor. Schedule an appointment with her doctor or go with her to her next doctor appointment. Discuss your concerns with her doctor and he/she will be able to evaluate by medical history and doing a dementia work up if indicated. Her doctor will also be able to refer to a neurologist and/or neuropsychologist for further evaluation/treatment as needed.
You can't, but. A psychiatrist should be able to help you figure it out.
Ask physician. A neuropsychologist will determine specific type of dementia. Sometimes, it can be mixed dementia. Geriatrician will help to collect information from her medical background and any diagnostic testing. Please look it up at www.Alz.Org and look for ten warning signs.