I have chased anything from general lightning producing storms to violent supercell (rotating thunderstorms) that produce giant hailstones and tornadoes. I have not only been close to these, I have been inside these monster sized supercells (not recommended).
A severe thunderstorm is defined as such because it is producing severe or even violent weather conditions that most people only experience realtively few times in their lifetime. When nearby a storm, anything from inflow to cold outflow, lightning and thunder can be experienced as well as occasional rain drops or even hailstones. SOme of the most awesome storm structures and features with great contrast can be observed from outside of the storm including storm scale or localised rotation and spectacular shelf cloud formations. If the target was correct and conditions favourable, tornadoes can be observed and are best experienced from a resonable distance.
When entering the core, the environment changes rapidly. Anything from violent winds, blnding rain with reduced visibility and even large to giant windscreen smashing hailstones can be experienced. Entering cores are dangerous way to chase and should be avoided at all costs. Furthermore, you may not see it but experience tornadoes. Making decisions in such reduced visibility is difficult amd may not allow sufficient time to make moves to safety.
I have also been within 100 metres of a strong tornado - an experience so eerie during the event as I felt afterwards! For most storm chasers, the tornado is the ultimate in storm chasing. Being up-close can be experience - that can turn nasty or deadly!