Firstly we need to consider the nature or feature of both rain and snow as this actually depends on the medium that can be used to protect oneself from their effect. It can be observed that when we consider the weight of rainfall and sunlight under extreme condition, we can detect that snow tends to have more weight than water and it also melts after sometime to form water.
The main reason for using an umbrella is to protect ourselves from environmental influence or unfavorable weather conditions or climate such as sunlight and rainfall. The main reason for this is because some people don't actually feel comfortable in such unfavorable climatic condition (rainfall and sunlight). This also explains the main reason why wearing of cloth is used as a means of protection against weather conditions.
To answer your question in summary, the main reason we use umbrella whenever rain is falling is because of the shape of the umbrella. The nature of water gives rise to flowing of water down the umbrella such that we feel protected from rainfall.
When it comes to snow, there is high tendency of the snow sticking to the surface of the umbrella which adds more the weight of the umbrella and it doesn't give rise to free flow unlike water unless it melts. This simply explains that if the snowfall is in excess, there is high chance of the umbrella getting covered by the snow unless it melts to form water for free flow.
Thanks for reading and I hope this helps.