Does "Connected Causes" mean the same thing as Dependent Origination?
In discussing things with people it helps to use the terminology they use. So even if 'connected causes' is a better way to explain 'dependent origination' - the latter has become a common phrase in English Buddhism. The former is rare. 

Without the precise quote it is impossible to determine whether the two are precisely the same thing. However, if the source is a biography, I would think that the meaning is likely close enough within that context. 


  
    
      edited in response to elaborated question:
    
  


In this case I would understand the phrase 'Connected Causes' as very simply all the causes that  create good circumstances (aka 'worldly good'). The sentence merely states that while it is great to experience the result of good karma, that doesn't mean you should rest on your laurels.

In other words: 'Connected Causes' is simply karma here, I think. And while the spiritual boon may seem the same thing as those great circumstances, it's not. For one thing: good circumstances are rarely used to good spiritual effect.