First off, racquet stringing is done by people, not machines. A stringing machine is a tool used by a racquet stringer to string a racquet. In your case, your old stringer appears to have been an accomplished technician. Chances are, even if you could get your hands on his old machine, the work it would produce would feel different to you with a new operator.
That said, different mounting systems and tensioning mechanisms do affect the output of a machine. If you have been getting your racquets strung on a Star 4 and then you switch to a stringer using a Neos 1000 you are going to feel a huge difference for a variety of reasons:
1. The Star 4 is a constant pull machine and the Neos is a lockout machine. The Star 4 will therefore produce work 5-10% tighter even if the machines are both calibrated perfectly.
2. The Star 4 has a six point mounting system and the Neos has a 2 pt mounting system. In my experience the 2 pt mount produces a stiffer string bed.
3. The Star 4 has diamond dusted clamps which do a superb job of preventing tension loss during stringing. Can make a huge difference especially with coreless multifilament strings like Xcel.
So, yes, machine brands make a difference and machine types make a difference. Still the single most influential variable you can change is the person doing the work.