How would society look different if the scientific method had not been developed?
There is no way to know this for sure. However, we can certainly speculate that society would be much less technologically advanced and perhaps much less structured as well. In other words, the scientific method may have affected our society both technologically and societally/culturally.
Technologically, it is likely that we would be much less advanced had the scientific method never been developed. This does not mean we would be without technology. Technology can be developed through trial and error or simple luck. People had already discovered things like herbal medicines, gunpowder, and some kinds of farm technology even before the scientific method was created. Even so, it is unlikely that our technology would have reached the heights that it has. Perhaps we would be at the same technological level as people in the 1600s or 1700s.
It is likely that our society and culture would be different as well. Our use of the scientific method has (one can argue) caused us to become very organized and (in Max Weber’s terms) rationalized. We try very hard to make our society as predictable as possible and we try to base it on rules. This has come about in part because of the rise of the scientific method. We want to apply that kind of intellectual rigor to more parts of our society instead of simply letting things depend on human nature and human emotions. Without the scientific method, then, our society and culture would probably be less rational, more oriented to human emotion, and less structured/organized.
In these ways, a society that had never experienced the rise of the scientific method might well look very different from our own.