What weather should we expect when temperature and dew point are similar?
Both 20/20 and 4/4 will probably result in fog or thick haze/mist.
4/4 is a problem because it tells you that besides the airport being likely fogged in, the freezing level is only a couple thousand feet above the airport and there is very likely icing at that level if there is any cloud up there. Only airplanes capable of flight in icing would be coming and going, if at all, depending on the actual runway visibility.
20/20 is also likely fogged in, but the freezing level is up at 10000 ft so that's not a problem.  15-20C is a sweet spot for carb icing, so saturated air at that temperature is a major concern.
The other thing to take into consideration is temperature tendency.  Generally, a zero spread like that in the morning means you can expect to wait a couple hours and the fog will lift.   On the other hand, a spread approaching zero in late afternoon is a big red flag.
When I was bush flying on floats in Northern Ontario, the dew point spread was the most important bit of data when checking weather.  A spread of one degree in the morning when departing on a trip was no worries because it would only improve (absent frontal conditions etc) as the day warmed up. On the other hand, a tight spread in the afternoon might mean making a customer wait at the lodge until the next morning rather than risk getting caught out scud running home at 500 ft, with clouds forming all around as the spread became zero, necessitating a diversion to the nearest lake on the route.