Technically a start up can be a traditional business, operating traditional models such a physical location for exchange of goods and services, so the difference is not too significant.
A start up simply refers to a new, emerging, fast growing company that aims to scale beyond the ownership of a single owner, using a scalable business model.
It can very well be a traditional business, but it is often used primarily in the rapidly expanding Digital World that is making use of innovations and technologies that do not require traditional business locations for transactions.
I think the biggest difference between a start up and a traditional business is in the vision. Some traditional business owners may be ok with their size as long as they are profitable, but a start up seeks growth beyond its current state, environment and ownership