In terms of differences, eukaryotic cells are much larger and more complex than prokaryotic cells. On average, a typical eukaryotic cell is about 1,000 X larger in volume than a typical prokaryotic cell.
This means both drugs (amoxicillin and penicillin) can treat a wide variety of infections. However, unlike amoxicillin and penicillin, the routinely prescribed drug, metronidazole, has a more selective action towards anaerobic bacteria. These bacteria are the ones causing the signs and symptoms of bacterial vaginosis.
Eventually, scientists learned that penicillin kills bacteria by inhibiting their cell wall synthesis. In other words, it stops bacteria from making a strong, correctly-formed outer wall. Nowadays, there are many other antibiotics besides penicillin that also target the bacterial cell wall.