While these terms are often used as synonyms there is actually a substantial and important distinction to be made between them. According to Wikipedia:
Conventional wisdom is a term used to describe ideas or explanations that are generally accepted as true by the public or by experts in a field.
Common sense is the basic level of practical knowledge and judgment that we all need to help us live in a reasonable and safe way
So, as we can see, something can be viewed as conventional wisdom if enough people believe it or if enough influential people claim it to be true. It does not have to be actually true. Common Sense refers to real knowledge that is based on something other than popular opinion. It has some real epistomological basis.
Common sense often flies in the face of conventional wisdom. People intuitively know that some things are true even if many of the so-called experts do not say so, and they find this very frustrating.
This is what our group, Seacoast Common Sense Community Builders is trying to promote and what I tried to convey in my book, Discovering Possibility: A Common Sense Conservative Manifesto: For Classical Liberals Too. Much of what has been passed off as truth for the past one hundred years or so is nothing more than conventional wisdom that has been sanctioned by influential people who had various political agendas. We believe there are some common sense ideas that used to be conventional wisdom too, but have been discarded by these social forces because the common sense ideas no longer fit their agendas. We want to rediscover these notions and pass them along in as many ways as we can.