Here's an extract from an email from one of the guys who worked at Solartron for a long time and>is still there now that it is Ametek:

Incidentally did you know that the 7081 and 7071 are essentially the same instrument?
The reference zener diodes were soaked for several weeks in an oven which temperature
cycled them and each diode was checked regularly for stability. The very best ones were
selected to go in the tighter specified 7081. The not so good ones went in the 7071.

However, I checked with the service guy here who does most of the
DVM repairs and he tells me the only difference is, as you suspected,
a link on one of the earthy logic boards, board 14 (or board 4 on older
models). There are a pair of solder pads next to SK414. LK17 is shorted
to make the instrument think it is a 7071. These links may not be present
on older models where the link was hard wired. In any event the object
is to make pin 2 of IC424 low for 7071, high for 7081.
This line is read on power up.

The drawings we've got here show this line connected to the "interface"
section on the back panel GPIB selection switch and indeed on the
prototype models it was, which meant they could be switched from 7071
to 81 by flicking this switch. This line was isolated from the switch
on production models.