With VSync on, yes, having a higher frame rate can be beneficial since VSync effectively "stores" up all frames generated since the last monitor refresh, then when the monitor is ready to refresh again, VSync hands over the most recent frame. However, you're really at the mercy of WHEN the last frame was generated. You can still get a frame thats only 1 ns old from 61FPS, it's just less likely to happen compared to a 120FPS frame rate. In fact, when you have VSync on, you really want to make sure that (ideally) you don't even have a minimum frame rate near 60. That's because, if the frame rate drops below 60fps (even a nanosecond less), your monitor is ready to update again, but there are no stored images since the last update, so the last image gets displayed
again (stutter). This can be very "bad".
To counteract the latency inherent in VSync when you get runt frames dropping below 60fps, having it off means there are multiple frames shown on each monitor update (tearing). Say you're generating 180fps, seemingly you'd have
about 3 partial frames displayed on each monitor refresh. Having a higher refresh rate here results in many frames being shown on the same monitor refresh, which
can reduce the severity of the tearing (positional difference) at the intersection, but it's still not ideal. Competitive gamers (before the introduction of variable refresh rate monitors) chose to live with tearing in order to always have the most recent frame being displayed on their screen (typically at the top since scans generally happen from top-down). However, regardless of how many frames you've generated on your monitor refresh, your oldest frame (from 16.7ms ago on a 60Hz monitor) is always at the bottom of the screen (so keep your head down!).
Example of tearing. Newest frame on top half, old frame on bottom half.
And here's what non-vsync tearing looks like in action using FCAT. The colored bars on the left side are the limits of each actual frame.
Both AMD and Nvidia have "adaptive VSync" options available in their drivers. When you go over 60fps, take VSync for smoother animation, when under 60fps, accept tearing to preserve animation smoothness.
I'd argue that with the cost of entry for variable refresh (FreeSync) monitors being comparably low ($200 or less for a 24" 1080p 144Hz FreeSync monitor) and with higher refresh rates coming down in price every month/year, it makes sense to upgrade if you want to be competitive. 60Hz fixed refresh monitors are arguably "low end" these days.
*To use FreeSync with an Nvidia GPU, you'll need a 10xx or 20xx series card and the monitor needs to have a DP input. Older cards will require a GSync display, which typically sell for a ~$150-200 premium over FreeSync monitors.