The biggest pro in separating personal from business savings is establishing clear evidence of separateness, which is an important presumption with LLCs or other types of business entities. As a refresher, one big reason entrepreneurs form LLCs is to limit their liability to their business investment. If you have an LLC but don't open a separate business account for that entity, then it's really hard to say with a straight face that you're operating as a separate business entity.

By contrast, there really aren't many cons to establishing a separate business account. I suppose you could consider "hassle" a con, but that's likely greatly outweighed by the benefits.