The change that effects the greatest number of people is that Ohio will now recognize any valid permit from any US state in Ohio. So if your permit was recently not honored in Ohio you now have some additional flexibility when you travel to the Buckeye State. This is also good for Ohio permit holders as Colorado and Georgia are returning the favor and will now honor the Ohio permit.
In addition to the greater reciprocity, the new law also allows hunters to use suppressors on guns, allows Ohioans to buy rifles, shotguns, and ammunition from any state; and impements a more rigorous background check for concealed carry permit applicants.
Also in good news the law decreases the number of required hours to obtain the permit down from 12 hours to 8 hours and allows that some (but not all) of that instruction take place via the internet.