Advantages of natural third stage (disadvantages of the injection):
You have longer to get the placenta out - a placenta trapped behind a closed cervix (which can happen once syntometrine is given) can contribute to PPH and therefore the risk of retained placenta and needing manual removal is reduced.
Fewer drugs in your system
No-one can take baby away for cleaning up/weighing etc as they're still attached to you
Baby gets full amount of cord blood
Disadvantages of a natural third stage (advantages of the injection)
Increased risk of PPH if your uterus doesn't contract efficiently once baby is born (although syntometrine can be given at any time, it doesn't have to be within seconds of the birth)
Shorter time to deliver placenta once injection is given so you're on a shorter clock although it is usually effective at delivering the placenta fairly quickly (however if it hasn't been delivered within so many minutes afterwards then you may need manual removal)
Breastfeeding immediately (if you can) and being left to bond with your baby are actually meant to help deliver the placenta - the oxytocin release helps the uterus to contract and encourages the placenta to shear off.
Some midwives aren't comfortable with physiological third stage or aren't aware that they shouldn't touch the cord or apply gentle traction to it without drugs.
If you're having a physiological third stage, the idea is to 'watch and wait' - some MWs don't seem to be entirely comfortable with that and feel that you 'should' be doing something to help things along (ie changing position and pushing), when actually the best thing to do is observe and then suggest the injection later if doing nothing doesn't work.
Prophylactic use of syntocinon (one of the drugs used) has been shown to be less effective when used in subsequent pregnancies (eg if you have it 'just in case' in one pregnancy, then it is less effective next time it is used) - I think that research was done by Professor Cecily Begley at Trinity College, Dublin. Her research also showed that there was no increased risk to mothers from watching and waiting and administering syntometrine if it was necessary as it still only takes a few seconds to get into the system.
I've looked into this a lot as I had a PPH and retained placenta first time round, and am planning a homebirth with physiological third stage this time round. The Supervisor of Midwives didn't disagree with anything I said to her about the above