Like anything else, it is one tool that can be used to try and predict activity levels. Deer do relate to changing light levels as that is one of the triggers for breeding activity. The tables will generally show that the 2 days before and after a full or new moon are very good for wildlife activity. Some feel that a new moon will stimulate more daytime activity.
You still need to factor in what your property is to deer. Is it transition to and from food, bedding, along with weather, etc.
I think local weather conditions have a far greater influence on deer than the moon. If all other things are equal then the moon may play a deciding factor, but I tend to focus more on wind direction, and weather fronts. I love to hunt before an approaching cold front (bigger the temp dif expected the better) and the first clear day after a nasty front has moved thru. I don't put much inventory into the solunar tables - just hunt the wind when I can hunt.
Like I assume a lot of other folks I was skeptical about the accuracy of the tables. 5 years ago I started watching the tables but hunted as I had always done and kept up with sightings in relation to what the tables said. The accuracy of the tables has been way higher than I expected, around 85-90 pct. still don't hunt "by the tables" because as others have mentioned there are several other factors that come into play such as weather conditions, rut phase, hunting pressure, location of stand in relation to bedding and or food, etc. The one thing that I observed and now try to account for as well is that deer are going to get up and move at least some every 3-4 hours so I note that time window on either side of projected major movement periods, esp when hunting close to bedding. I have the scout look app on my phone.