In standard form, there is nowt in it in terms of straight line speed.
350 is more raw, whereas the 370 is more refined - but both are definitely 'drivers cars'.
As the HR 350 and 370 use the same base engine both are far more tuneable to 500bhp without the need for forgeing the internals, unlike the 276/296bhp 350s where the max safe bhp is about 440bhp (driving style dependent ).
The 370 auto option appeals to many and appears to be more bullet proof than the manual boxes.
I loved my 276 350 but missed the torque with the 296 rev up 350 and didn't appreciate the increasing oil consumption those engines are known for. The 370 with an uprev is very satisfying both in overall speed and torque, for me.
The icing on the cake with the 370 is the synchro rev - but the OEM exhaust is dire in terms of sound quality so when a decent exhaust is fitted you will find your self making additional gear changes just to enjoy the synchro rev.