North-hemisphere ice-shelf melts on summer and grows on winter. I would expect appreciable changes on sea-level between seasons, but sea-level looks equal on winter than on summer.
Why doesn't sea level show seasonality?
Sea level has a strong seasonal signal. The annual variability is less than the daily changes associated with tidal forcing in most locations, but still can be on the order of 5-10 cm (maximum values about 15 cm).
The causes of the seasonal fluctuations are mostly associated with seasonal changes in wind intensity and patterns, changes in temperature that relate to thermal expansion, and in salinity (haline contraction) and river discharge fluctuations. The annual sea level cycle is only partially related to ice melt and this effect tends to be quite local.
The largest sea level seasonal cycles are associated with areas in the vicinity of large rivers with strong seasonal cycles (e.g., Bay of Bengal). Also, there is a lot of spatial variability in the seasonal cycle with the northern hemisphere having a larger signal (likely caused by the stronger seasonality in wind patterns).
An example of the seasonal cycle can be seen in the monthly data from the Permanent Service for Mean Sea Level (PSMSL) for Woods Hole, MA (USA) (in m offset to avoid negative values in the PSMSL database). The monthly data shows strong seasonal variability and also a clear trend. As the data is monthly averaged, the tidal oscillations are filtered out. Most sea level rise graphs tend to use annual data and thus the seasonal information is not included.
In the Arctic there is mostly floating sea ice and the mass of this does not change the sea level.
When snow falls on the floating sea ice the weight of the snow is the same as the weight of the sea water it displaces so there will not be any change in sea level from this.
But the snow falling on land during winter is water temporary removed from the sea, so the sea level will drop a little during winter. See seasonal changes in sea level.
In the Antactic winter, the sea level drop will be less as most of the snow will fall on the floating ice and on a narrow band of land ice around Antarctica. This is mainly due to the circumpolar wind around Antarcica where less moist air will blow into the interior of Antarctica.
The tiny bit of snow falling in the central part of Antarctica is called diamond dust, this is tiny flat ice crystalls; they will sublimate before or soon after they reach the ground due to the dry air and low temperature.