How to write a simple feedback loop in difference equation?
The first transfer function type you mention is the continuous-time Laplace transfer function.  This is a function of s where s=jw (can someone give this some LaTeX love?).  The difference equation form you mention is for a discrete-time system.  If you don't understand the difference or the terms I'm using then you need to do more background research into signals and systems.  

If that makes sense, then the basic way to convert between the two types of system is to use a Z-transform.  The simplest is to substitute a function of z for each s in your continuous-time transfer function.  Wikipedia describes it here.  Using the Bilinear Z-Transform the equivalence is (again, more mathy love please!): 

s = (2/T)*((z-1)/(z+1))

where T is the sampling time of the discrete-time system.

Keep in mind that there are other transforms as well.  Each has different uses and strengths.  This is very very basic.