I don't think it is possible for anyone to really answer your question, OP, just because all grammar schools are not identical nor all private schools. Some grammar schools (eg my dd's) are very good at stretching the most able, providing a wide range of extra-curricular activities etc. Some are less so. Likewise, some top private schools provide a particularly high standard of education, others don't. All those arguing about 'grammar schools' or 'private schools' above are only basing their judgement on their knowledge of specific local schools. Without knowing the 2 specific schools you refer to, none of us can compare them and really give you a definitive answer.
As the mother of 3, with the eldest at grammar (private wasn't an option for us financially but i prefer state anyway), i would have been wary of giving eldest an education that I struggled to afford and couldn't offer the other two. As others have said, this will allow you to have the cash to pay for all the other activities/support for all 3 not only to benefit your eldest.
I think there are some valid generalisations that can be made about grammar v private though - whilst one can't generalise on the academic attainment, it is possible to generalise in some areas that may help to clarify your decision-making process. Firstly, grammar schools are comprised of people who get in on merit only not cash, so they tend to have a different social mix and different values. Your dd is less likely to be surrounded by kids entirely from v wealthy homes, so less likely to have huge pressure about living in the 'right' type of house, driving the 'right' car and crucially for teenagers, wearing the 'right' clothes (well, that may still happen, but the 'right clothes may be from Atop Shop, say, rather than Armani or something). I would hate to be the poorest child in a school. Plus the grammar school ethos matches my own personal politics better. Though if you are Tory you might feel right at home in the private school.
claig has a point, to a degree, in that I think private schools probably are slightly better at instilling confidence, and yes, more of your dd's friends there would be likely to be loaded and possibly useful connections in later life. But does that matter to you?
Other considerations - grammar schools do NOT charge fees of any descriptions and are not allowed to, contrary to what is stated above. They are allowed to ask for voluntary donations and they are just that - at my dd's school the head is always moaning that only 30% of parents pay the charge. Plus university admissions are probably similar for identical candidates - but top private schools probably push more of their pupils to apply for better unis in the first place, whereas at grammars the pupils are maybe less likely to apply (the confidence thing again? or different attitudes from the school or parents?).
So lots to consider - for an accurate answer you'd need to name the schools! Or better still, speak to as many parent/kid at these schools as possible and see what they think.