Results on $\phi $ meson production in inelastic $p + p$ collisions at CERN SPS energies are presented. They are derived from data collected by the NA61/SHINE fixed target experiment, by means of invariant mass spectra fits in the $\phi \rightarrow K^+ K^- $ decay channel. They include the first ever measured double differential spectra of $\phi $ mesons as a function of rapidity $y$ and transverse momentum $p_{\text {T}}$ for proton beam momenta of $80 \, \hbox {GeV}/c$ and $158 \, \hbox {GeV}/c$, as well as single differential spectra of $y$ or $p_{\text {T}}$ for beam momentum of $40 \, \hbox {GeV}/c$. The corresponding total $\phi $ yields per inelastic $p + p$ event are obtained. These results are compared with existing data on $\phi $ meson production in $p + p$ collisions. The comparison shows consistency but superior accuracy of the present measurements. The emission of $\phi $ mesons in $p + p$ reactions is confronted with that occurring in $\mathrm{Pb} + \mathrm{Pb}$ collisions, and the experimental results are compared with model predictions. It appears that none of the considered models can properly describe all the experimental observations.