Ok, I'm going to come at this from a different angle, having gone through both systems with very different boys.
I agree with ChocolateWombat that CE is pretty much irrelevant, as pretty much all boys will reach the required CE scores to confirm their conditional offer. BUT in our experience 13+ and 11+ have been very different in terms of the competition for the initial places in Year 6.
In short: 11+ allows you to bypass CE, giving the child a more relaxed time in Yrs 7-8, provided that they're ready to progress to secondary school at the end of Yr6.
13+ on the other hand gives you the security of still being at prep school - a big fish in a small pond - for another 2 years, before moving on to senior school, potentially much further away.
For our DS1, 13+ was obviously the right choice. Highly academic, he had a choice of top schools after the Yr 6 pre-tests. He decided to sit the scholarship exams rather than CE, and this was hugely stimulating. He had a great time as a Senior in his prep school and loved being in the "scholarship class".
DS2, a couple of years younger, was a totally different case. We felt that he was more mature and ready to move to secondary school for Year 7. He was also fed up with exams and mock tests, and we were worried about how to motivate him for CE. So he sat 11+ rather than 13+ pre-tests.
In terms of competition, there is no comparison in our experience: 11+ was BRUTAL compared to 13+. This may be partly due to the fact that DS2 sat 11+ three years after DS1 did his pre-tests and it's clearly getting harder year by year. All the schools told us that numbers recenly increased dramatically. One went from 600 candidates to 1,000 in just one year.
BUT we heard from other parents in DS2's year who attempted a mix of 11+ and 13+ exams, and who reported that the 11+ experience was depressingly anonymous, with candidates, with parents queuing for huge distances before the exams, whereas at 13+ they were still treated like individuals, just like we experienced with DS1.
I appreciate that this is still very subjective, especially as our DS1 was scholarship material and so was invited to sit 13+ exams in a very exclusive environment. But nevertheless, our DS2 (who is somewhere in the 2nd quarter of children by CAT scores, according to his school) found the South West London 11+ process absolutely gruelling.
Now that DS2 has secured an 11+ place for September, I'm delighted that we chose this route for him and he won't have to sit Common Entrance. Knowing how stressful this past term was for him and for us, I'd probably choose 13+ next time as his school friends who are similar ability all got places at 13+ for schools that rejected or waitlisted him at 11+.