There haven't been trials directly comparing wine to grape juice (outside of studies in animals such as hamsters), so the comparitive benefits are theoretical. Grape juice is different enough from grapes, due to lack of fiber and consequent effects on gut bacteria and other factors, that the results of whole grape trials aren't transferable to grape juice.
Wine contains alcohol (duh!), and alcohol in moderate amounts may help relax blood vessels and increase HDL. Wine also has higher amounts of certain phytochemicals than grape juice does, but similar amounts of other ones.
It doesn't appear that grape juice reduces blood sugar, insulin, blood pressure, or improved lipids. However, one study that used a highly concentrated juice was more likely to find benefits, so perhaps it's a dosing issue. There are a variety of studies, largely sponsored by a large juice company, that have explored other areas of health, and the results are so far inconclusive.
It's important to note that there are very very few trials on grape juice, compared to on wine. So the reliability of the total body of evidence is somewhat low.
Grape juice doesn't contain alcohol. Consuming alcohol in moderate amounts can turn into consuming alcohol in large amounts, whether chronically or even on occasion.
Even in moderate amounts over time, alcohol may not be that healthy. Some researchers believe the evidence for benefits is mixed enough, and potential detriment large enough, that health benefits should not be trumpeted.
Grape juice however, and fruit juices in general, are linked to tooth erosion and dental carries. However, the trials on this typically use conditions that don't fully reflect real-life juice drinking. Grape juice in amounts that are clinically effective may also contribute excessive calories into your diet, as juices are much easier to consume than whole fruits.
Often but not always purple grape juice, at 100% purity (in other words, not mixed fruit juice, or sugar-added juice). Occasionally, white grape juice is used.
In general, sufficient trials haven't been conducted which compare different fruit juices against each other. There are a limited number of studies, such as one finding no difference between grape juice and apple juice and one finding benefits of grape juice over orange or grapefruit juice. But these were for highly specific outcomes rather than practical health benefits. Much more research is needed. We'd love to know how juices compare, but the financial incentive isn't there for companies to fund studies pitting juices against each other.
All that being said, grape juice doesn't appear to help memory and cognition quite like blueberry juice can. Unfortunately for the consumer, blueberry juice is much more expensive than grape juice.