  We find, at the Lagrangian off-shell level, the explicit equivalence
transformation which relates the conformal mechanics of De Alfaro, Fubini and
Furlan to the conformal mechanics describing the radial motion of the charged
massive particle in the Bertotti-Robinson AdS$_2\times S^2$ background. Thus we
demonstrate the classical equivalence of these two systems which are usually
regarded as essentially different ``old'' and ``new'' conformal mechanics
models. We also construct a similar transformation for N=2, $SU(1,1|1)$
superconformal mechanics in N=2 superfield formulation. Performing this
transformation in the action of the N=2 superconformal mechanics, we find an
off-shell superfield action of N=2 superextension of Bertotti-Robinson
particle. Such an action has not been given before. We show its on-shell
equivalence to the AdS$_2$ superparticle action derived from the spontaneous
partial breaking of $SU(1,1|1)$ superconformal symmetry treated as the N=2
AdS$_2$ supersymmetry.
