January 3rd was an interesting date in orbital mechanics; it represented the day when the Earth was closest to the sun (perihelion) and the moon was farthest from the Earth (apogee). The first only happens once per year; the second happens every lunar orbit - every 29 days. These two effects have a lot of star watching history.
The Earth orbits the sun, with one orbit being one year. While it orbits the sun, it spins on its axis, and each spin on the axis makes for a day. While the Earth rotates around the sun at a bit over 30 kilometers per second (108,000 km per hour!), and is spinning like a top (at around 0.45 km/second) you or I don't feel this motion - though it is noticeable because the combination of these two motions sets the yearly procession of the constellations and the day-night cycle...and is why some telescopes have clock drives to keep them pointed at a target once it's been set.
What do perihelion and lunar apogee mean to you? Well, when it comes to observation, the difference is a function of the eccentricity of the orbits. Eccentricity, if it's been a while since your algebra class, is the ratio that defines the distance between two focal points of an ellipse. For an orbit, eccentricity will always be between 0 and 1, anything of 1 or more isn't an orbit - it's a object flying through the solar system on a hyperbolic path.
Earth's orbit has an eccentricity of 0.0167, which means that the difference between closest approach to the sun and farthest approach to the sun is 0.0167% of the mean distance from the Earth to the Sun. The Moon's orbit is somewhat more eccentric, at 0.05.
From an observational standpoint, the difference in visual size between perihelion and aphelion (the farthest distance from the Earth to the Sun) is minimal; the average difference between the Earth and the Sun is 500 light seconds, or about 150,000,000 km. 0.0167% of that is about 2.5 million km of difference below the average, and aphelion is about 2.5 million km farther away. Because the sun is so far away, that variation in distance, as far as it seems to us, makes almost no difference in the visual size of the sun. For the moon, the difference translates into something that would be a noticeable difference in angular size - if the events didn't happen 15 days apart in a different phase of the moon. If you can catch a photograph of the full moon at perigee and apogee on different occasions, you'll notice that the moon appears almost 10% larger - about 4.1 arc seconds larger at perigee than apogee.
The secondary effect of the lunar apogee/perigee combination is a difference in lunar tides, as tidal attraction works on the cube of the difference in forces. This is the variation in orbital mechanics that has the greatest impact on human society.
From a standpoint of living on Earth, one of the oddities of Earth's perihelion is when it occurs. It occurs near the height of the summer for us (January 3rd), which means it's in the depth of the winter for the Northern hemisphere. There is a very small difference in total solar radiation (called instellation) of about 4% between January and July - though it does mean that seasons at aphelion are about 4 days longer due to Keplerian mechanics.
That may not sound like much, but it's significant due to the time of the year those differences come in compared to the seasons, and how this changes over time. Right now, the current cycle of perihelion and aphelion moderate Northern hemisphere summers and winters; over a 100,000 year cycle, the seasons that perihelion and aphelion occur in shift; these shifts are called Milankovic cycles, and the last time aphelion occurred during the Northern hemisphere's winter, we were in an ice age.. Some of the hottest climates in geological records appear to have occurred when the Earth's orbit was a bit more eccentric (variations in orbital eccentricity are also called Milankovic cycles), and perihelion and aphelion happened closer to the equinoxes.
This isn't to say that there aren't other contributing causes to variation in the Earth's climate, nor should we discount the effect of CO2 in the atmosphere. It is pointing out that there are LOTS of contributing causes, and some of them (the Earth's orbit) are well beyond our control.
So is the earth hotter in the southern hemisphere or the winters milder in the northern hemisphere due to Perihelion falling in January ? The answer is yes, but no more so than in your grandparents day or their great great great ... grandparents!