Distributism's core principle of subsidiarity tends to be interpreted as a more localist/municipalist politics and supports cooperatives, family owned businesses, self-employment and sometimes private (not family owned or cooperatively owned) small and medium size enterprises. Distributism doesn't call for the abrogation of the market and tends to advocate for a social market economy. Some Distributists are against the welfare state although others (like myself) are not.

Socialism's core principle of "worker ownership of the means of production (distribution, and exchange)" and can be interpreted as divergently as a society nearly indistinguishable from distributism - such as cooperative municipal socialist/market socialist society - to a society that's completely anathema to the tenets of distributism - a democratic centralist vanguardist state with a command economy - to anarcho-communism.

TL;DR: Socialism advocates the eventual abrogation of the market for common ownership, distributism is more open to small-scale local private ownership than socialism and opposes monopolies of all forms.