Does the maternal Grandmother have more clout?

I think it depends on all sorts of things. Personalities, proximity, time available to spend helping or visiting etc. and how many other grandparents/step gps are involved.
I'm not competitive at all and I just join in when I'm asked and do what I can. I love them all and they know it,and so far it works.

I think it's natural for girls to turn to their mother (in most situations). It's what they've been doing all their lives.

I am far more involved in my DGCs lives than their paternal grandparents.

With DD1 her in-laws are divorced and remarried & so not sure if this made a difference as they are with partners not directly related to DGs (though all love him!) They live close (15 mins away) and he does visit them at least monthly and sometimes more..but it tends to be more 'formal' ie planned lunches, sleepovers etc. Wheras I look after DGS 2 x week plus DD will pop over at weekends if her husband is working or ask if I can look after him for short periods whilst she goes to supermarket etc. I suppose I tend to prioritise DDs support needs as my mum did for me. Am not a 'doormat' but try to help if I can!Feels like other grandmother loves our DGS but does not always offer support but waits to be asked. Feels like she does all on her terms,

DD1 partner has poor (almost non-existent) relationship with his mother & she has never seen them. She met my DD2 when she was 5 months pregnant with her first DD but MiL chose not to make further contact! Apparently she is close to her own DD and is v involved with her family. She lives in midlands & DD2 is South

I do think it's only natural that daughters tend to be closer to their own mothers, and sometimes there can be female rivalry (even if unadmitted) between the mother and the wife of a son.
A lot depends on the individual relationships between the parties. Sometimes the DIL is virtually a stranger, and other times she's been a welcome family member for a long time. Personalities can clash, too!

I don't know about clout, but I certainly find I can speak more openly with daughters about their babies than with ex-DIL (mind, she wasn't your average DIL). I find with both daughters that I can make suggestions if they ask for advice and they won't necessarily act on it till further down the line, when it becomes their idea! My daughters make suggestions to me nowadays and try to tell me what I should be doing decor-wise or what gadgets I should buy. I sometimes listen - same as them!