I came across this circuit for a non inverting op amp which is biased Vs/2. I'm wondering what the difference bettween these two circuits are? What is the purpose of R3 in the first circuit and how is it different to the second circuit?
They're both the same. The first circuit assumes a Vbias supply is available independently. The second provides it from Vs as a Thevenin equivalent. In this case, R4||R5 = R3, and Vbias = Vs*R5/(R4+R5) if the circuits are to be functionally identical.
In case R3=R4||R5=R3/2 both circuits provide the same gain - however, the first circuit uses dual power supplies (+ - Vs) and can work with Vbias=0 (ground), whereas the second one can work with one supply voltage only. Therefore, the second one needs a DC bias voltage at the non-inv. input for biasing the output DC at Vs/2 (R4=R5).
As a consequence, the input resistance of the 2nd amplifier is reduced to R4||R5.