The primary value of a screened bottom board is for ventilation. This can be helpful in high temperature and high humidity environments. Also, for some beekeepers screened bottom boards are a part of their pest management strategies. They can be used as a natural way to control small hive beetles and Varroa mites by putting a sticky substance on the screen to catch them as (if) they fall through.
While the added ventilation can be helpful in certain areas and certain times of year, there are some issues. Too much ventilation and/or moisture rising from the ground can make the hive harder to cool in warm climates. If there is too much ventilation in the hive it can impact communication due to queen pheromones escaping through the screened bottom board. Also the hive may be harder to keep warm during cold months. Some beekeepers keep the insert partially installed allowing for some ventilation during cold temperatures.
Bees are constantly cleaning their hives. While the use of the screen allows debris, mites and beetles to fall through the screen, it can collect on the board insert and the bees cannot get to it to clean. Therefore it is important to monitor the sticky board regularly when the insert is in place and keep it clean.
Caution: If you are installing a package of bees in a hive with a screened bottom board you must consider Queens do not like light. Keep in mind a hive is dark. So it is critical that you make sure your screened bottom board has the sticky board or tray inserted when installing a new package. It is common for the queen to abscond after a few days if there is too much light in the hive. That is why we only use solid bottom boards for our NUCs. So, make sure the insert is in place until the queen has started to lay (typically 10-14 days after installation). Once you have brood you can remove the insert.
As with any subject in beekeeping, ask more than one beekeeper and you will get multiple answers. And each answer can be right for them. We use both screened and solid bottom boards but we prefer the solid bottom board for most applications.