Title says it all really, just a simple experiment I'm doing.

Yes, it will. And a lot. Because the higher arc voltage of the overloaded tube will cause higher power transferred from the ballast to the lamp.
And very likely the lamp will be way more prone to cycling (higher arc voltage will make the ballast OCV margin not sufficient for a reliable stable arc).

70W and 100W HPS lamps in the US, have both 60V lamp voltage.

Yes, it will. And a lot. Because the higher arc voltage of the overloaded tube will cause higher power transferred from the ballast to the lamp.
And very likely the lamp will be way more prone to cycling (higher arc voltage will make the ballast OCV margin not sufficient for a reliable stable arc).

I actually tried this out last night and it only glowed grey with the arc stream snaking inside.

"Both lamps are 60V" - well yes, but that means the voltage drop across the arc is the same 60V (assume the arctube temperature so pressure remains constant).
Don't forget, with discharges the arc voltage is dictated by the lamp and the current by the power supply (the ballast). So when both 70W and 100W lamps have the same arc voltage, the same ballast will feed them by the current (so power) corresponding to the ballast design, so in your case it will feed both lamps by the same 100W. While the 100W lamp is likely designed to handle that power (obviously), for the 70W rated one it would be quite an overload. And HPS respond to an overload by elevating the arc voltage, which will likely lead to further increase of the real power delivered to the lamp (ballast is somehow maintaining the current, so higher voltage with the same current means higher power), easily in the 110W ballpark. So there we are talking already about 40% power overload of the lamp.

"It just looked gray" - well that means the lamp has not ignited for some reason.
It could be you actually had an European 90V arc voltage 70W HPS, which won't ignite on the lower voltage US ballast, The reason is, the European HPS needs at least about 200V OCV (the voltage present at the lamp terminals when no lamp is present, disregarding the ignition spikes) to start and run, while the US ballasts have just 120V OCV (for these power levels both US,  as well as Euro- ballasts are a series reactor coil, so the OCV is equal to the mains feeding the ballast). The ignitor may have caused a gas breakdown (hence the gray glow from the buffer gas as the sodium is still solid in the reservoir when cold), but there is not enough voltage present to turn the tiny initial spark into a fat high current arc.
Or the lamp you tried is just faulty, e.g. with a detached electrode, so the buffer gas gets excited by just the high frequency/high voltage pulses from the ignitor via some capacitive coupling. But without the decent electrode the thick fat arc won't form.

Okay, so the reason it didn't strike an arc was because I had forgot to connect the COMMON wire from the lamp shell to the other COMMON wires and now it ignites and gets brighter than it did at 70 watts.