"Does your university being a member of The Russell Group really make that much difference to a PhD student?"
Well RG universities tend to have the best international reputations, the most funding, and, consequently, should offer the best facilities and best academics. Basically, it's just the traditional universities defining themselves as the best. The press are dubbing the RG as the British Ivy League, and therefore, it is gaining a growing reputation of being the creme de la creme of British education. Although this has now been qualified again by the 'Golden Triangle' (Oxbridge, Imperial, LSE, KCL, UCL). One wonders who sits down and comes up with these categorisations!!