Hi,
The solar wind is a constant stream of particles (ions and electrons)
coming from the Sun. It is pretty much always there although it
varies in speed.
A solar storm is an large explosion of material from the Sun which goes
out into space in addition to the solar wind. There is a lot more
material leaving the Sun in a solar storm, which is part of the reason
they can be problematic for astronauts and instruments or satellites in
space. Solar storms occur occasionally, not all the time like the
solar wind. They occur more often when there are lots of sunspots
on the Sun.
Best,
Christina