This will be an ordinary prejudice case. To win you need to show that the disadvantage to your son from not being admitted outweighs the problems the school will face through having to cope with an additional pupil. This must be about things that will affect your son, not things that are inconvenient for you. Transport and child care difficulties do not win appeals. You must remember that you are appealing for this school, not against the current school.
I have to disagree with most of Charmatt's post.
Having a sibling at the school is not relevant for appeals. Mention it but only as a fact. It is not an argument for your older son to attend this school.
Drop off and pick up being more difficult is completely irrelevant. That is a problem for you. It is not a problem for your son. Appeal panels are told very clearly that logistics issues such as this are not relevant for appeals. It only becomes relevant if your son's journey is more than 45 minutes each way or if he has mobility problems that make the journey difficult.
Emotional, social and health needs will only be given weight if backed up by a letter from a medical professional stating that, in their view, your child needs to go to the appeal school. Without that evidence the appeal panel is unlikely to give any weight to such arguments. Exceptional needs of others in the family are not relevant at all.
Being the closest school is not relevant. The fact it is your preference is obvious - you wouldn't be appealing otherwise! However, it is not relevant.
Do not mention any reasons why you feel the other school is unsuitable. The only exception to this is if you have proof that your son is being bullied and his current school is failing to take adequate action to deal with it. You don't mention bullying so I presume that is not true. As I said previously, you are appealing for this school, not against the current school. At best the panel will ignore anything you say about the current school. At worst you may antagonise them - whilst none of the panel will be connected with the appeal school, they may be connected with your son's current school.
Concentrate on looking for things that the appeal school offers that are not available at your son's current school and that are particularly relevant to him. So, for example, if your son is musically talented and the appeal school has a lot of extra-curricular musical activities that are not available at his current school, that is worth bringing up.