Evidently the old $\phi$ is measured from an angle oriented $180^\circ$ relative to the new convention. This implies that the amide H ($H_N$) rather than carbonyl carbon was used as the reference fourth atom used to define the dihedral, since these atoms are related by a $180^\circ$ rotation. The old choice of atom is opposite on the chain to the modern convention of using the heavy backbone atom (carbonyl carbon). You can obtain angles in the new convention by subtracting $180^\circ$ ie $\phi \rightarrow \phi - 180^\circ$.
The meaning of angles with value of zero for $\phi$' aka $\psi$ is the same. However the values between $180^\circ$ and $360^\circ$ in the original Ramachandran plot can be shifted below zero, as $180^\circ\rightarrow -180^\circ$ and $360^\circ\rightarrow 0^\circ$, or more generally $\psi \rightarrow \psi - 360^\circ$.
By the way inspection shows that this is the correct interpretation. See for instance the following diagram. The extended (beta/turn/pII) region is usually larger than the alpha region. The beta region sits at large values (below $180^\circ$) of $\psi$, whereas the narrower alpha region is close to $0^\circ$ (at negative angles):
In Ref. 1 Ramakrishnan and Ramachandran explain the differences in the conventions:
At a recent conference of some of the representative workers in this field held in Bethesda, it was decided to denote the two dihedral angles about the bonds N-Ca and C.-C' by $\phi$ and $\psi$ respectively, the sense of rotation being the same as that adopted here. The fully extended chain, with N-H and C'=O trans with respect to one another, is to be taken as the standard conformation with $\phi$=$\psi$=0.
It is readily seen that the new $\phi$ is the same as the old $\phi$, but that $\psi$=$180^\circ$+$\phi'$ (and $\phi'$=$180^\circ$+$\psi$). All the data reported here are thus readily converted into the ($\phi$,$\psi$) coordinates. The diagrams in Figs. 2, 3, and 6 have to be shifted up by $180^\circ$, or half the total length, along the vertical direction. In particular, the right-and left-handed alpha helices will have ($\phi$,$\psi$) equal to ($133^\circ$,$123^\circ$) and ($227^\circ$,$123^\circ$). As before, a helix with ($-\phi$,$-\psi$) will be inverse to one with ($\phi$,$\psi$); i.e., it will be of opposite sense, but having the same number of turns per unit.
This paper was finalized well before this meeting, and so the older conventions are adopted here. However, it is proposed to use the new notation in the following papers in this series.
1.Ramakrishnan C, Ramachandran GN. Stereochemical criteria for polypeptide and protein chain conformations II. Allowed conformations for a pair of peptide units. Biophysical Journal, 01 Nov 1965, 5(6):909-933. DOI: 10.1016/s0006-3495(65)86759-5.