The answer to the OP question
What is the relationship between those two groups (are they the same or there is any subtle difference?)?
is offered by Niederer, U. in his Helv.Phys.Acta article called "The connection between the Schrödinger group and the conformal group", Vol. 47, p.119 (found at http://dx.doi.org/10.5169/seals-114561 and freely download-able). From this paper you find the reference  which is another "bite" (same journal, but Vol. 46) at this topic by the famous Asim Barut:
"Conformal Group -> Schrödinger Group -> Dynamical Group -
The Maximal Kinematical Group of the Massive Schrödinger
(the the direct link is http://doi.org/10.5169/seals-114492).
I would first read Barut's work, then Niederer's.