These two strollers are very similar in features, so I'll concentrate on the biggest differences. Btw, we have a full, in-depth stroller review of the Babystyle Oyster 3 HERE, so read it to know more.
Between the two, the Oyster 3 has definitely better wheels, chunkier, especially at the front. It also stands out by the superb padding and a headrest pillow, so the baby is comfier in the seat. Lastly, the hood, it is more open, especially in the laying position, BUT it is height-adjustable which means more adjustability and shading as needed. Also, the basket offers a bit more space here.
The Balios S is simpler, less padded (but enough), and a bit more spacious because of the more open seat (however, the Oyster has a longer leg rest). The wheels are also simpler, better for smoother surfaces, but it can go over grass if necessary. The fold of both is very similar, so none is bulkier really, and the Oyster is just slightly heavier - though the weight is very comparable. The price of the Balios, even if differing from country to country, is usually more affordable, so that is also a decisive factor (but I believe the quality of the frame reflects this, and the Oyster's is more durable in the long term).
All in all, these two are very similar in functions, and you should probably just think about which you like better, or simply go see them in real life to choose what your heart is telling you. If the budget is an issue, take that into consideration as well.