Why are there both climate regions from axial tilt and weather cells?
Generally speaking, the tropics are a geographical term. It's the area on the surface of the planet where at least once a year the Sun passes directly overhead.
It has relation to climate, but even then within the tropical region you have a variety of climate zones from rain-forests all the way to savannas and deserts. Being in tropical region only ensures that it's generally warm all year round, other parameters define the actual climate type. Take a look here at the range of climate types available in the tropics.
Now, coming to Hadley cells and how they relate to tropics - they are not directly related to one another, i.e. one of them is not used to define the other.
In Hadley cell the humid air rises and moves north or south depending whether it started north or south of the equator and then sinks to the surface at around 30 degrees north and south of equator. So they are both dependent on the same factors, namely the simple fact that it's generally hot in the tropical region.
What happens between 23 and 30 degrees really depends on where in the world you are. If you're between 23 and 30 degrees in North Africa, it's very unpleasant, because Sahara is right there. Other places are different depending on their own geographical and climate parameters.