The main difference is that the steam juicer, will actually concentrate some of the juice due to the hot nature of steam, which a LOT of it escapes.
As well, the juice you get is a gentle juice. I think for grapes, the press will actually work best, but I get great color extraction on grapes too with steam. The problem with the mechanical juicers, is that it only really works on fruit without seeds for wine. There would be no way you could do strawberry, blueberry, raspberry, etc with a mechanical juicer because it would also take those seeds and crush them up. Usually, the seeds bring bad tannins, off flavors, etc. You want to keep those seeds intact. Where the mechanical juicers really come into play is with very hard veggies and fruits. Apples work well in those mechanical juicers, as do celery, carrots, cucumber, etc.
But for berry fruit, steam juicers are awesome!