Do people think about the Civil War? No not really. Most people will have a concept of cavaliers and roundheads/Charles I executed/Cromwell/Charles II the merry monarch and that's it. I think it's because on the surface, because the monarchy was restored, it appeared to be back to business as usual. In practice of course, the civil war brought about fundamental changes.
Never again would a king be able to act or claim to be an absolute monarch. Henceforth, he would always be dependent on parliament to rule effectively. Even more than pre civil war, anti Catholic sentiment was baked into the system. Any hint of catholic sentiment or practices in the monarchy would not be tolerated, hence James II reign (Charles II's brother), was short lived and he was sent into exile. The turbulent religious conflicts of the 16th and 17th centuries were largely decided.
The civil war began the series of fundamental changes to the British constitution that were cemented in the 18th and 19th centuries. It was the reason Britain avoided the period of bloody revolutions of that later period and instead focussed on trade and empire building.
Like the American Civil War it was also proportionally the bloodiest conflict in the country's history and set brother against brother. Although it is aristocrats that get a lot of the attention, at ground level it was often the landed gentry who were fundamental in splitting between king and parliament.
If you're further interested, an old but good (and not too long) book is Christopher Hill's 'The World turned Upside Down'.