I'm pretty sure there isn't enough information in your question to answer this well, but I'll do my best.

Private schools are (generally) expensive, have small class sizes, have some kind of school philosophy (unclear what it is in this case, other than perhaps "privilege"), may have some different/interesting courses and/or field trips, and may or may not be a good fit for a particular student.

Connections is a program in a well-regarded public school, with more open-ended classes and discussions, and in 7th/8th applies just to english/social, so just a small fraction of the class time. To the extent it's different from "standard" JLS, it's probably best for kids who like to ask questions, and are more engaged than passive about their education. But I expect you would consider your child to be highly engaged, so that likely doesn't rule them out.

Based on your note, I'd suggest you look at the private school rather than Connections. I think you will like it better, and will fit in better, and they have (presumably) already determined it's a good fit for your child.

People told us Connections was only good for 6th grade, but we didn't listen, unfortunately. It was an uneven experience in that regard. Palo Alto in general has a nice parent comunity, and there are a few standout teachers. If your child has special needs of any kind, even giftedness, I wouldn't choose JLS. PAUSD in general is downright passive aggressive, and sometimes overtly aggressive to people who might need a little extra thought and work. Also, the classes were both not very challenging in core subjects but also too much busywork homework. If your child needs executive function or time to mature re organization, do whatever it takes to find something else.

I would try to take a tour of both. We felt lucky to be in Connections before attending, and there were definitely special things about 6th, but in hindsight wish we'd had the courage to seek something else when people said the middle schools here aren't so good.

Having transferred our son from the dreadful Jordan to JLS, I can honestly say that while JLS and Connections are superior to Jordan, it seems that ALL middle schools in PAUSD are rather substandard. This is why those who can afford to often take their kids out of the district for three years, going back to Paly or Gunn for high school.

However, some private schools ( Harker for example) are even more rigorous than PAUSD. Even Harker, Hillview and Bellarmine have had suicides-- but because they were not committed in public, they haven't had the negative press that Gunn has.

That said, most private schools are superior to PAUSD in our experience, and we found that colleges prefer to see a private school education, in most cases, to a public school education.

Most importantly, private schools give parents superior bragging rights.

I have heard good things about Connections but have not been part of it. It also depends on what your plans are for high school. I think lifestyle is important -- do you really want your kid commuting to San Jose? If so, go for it but I agree about what people say about Harker, I would say it has a reputation for being rather hard core. Imagine what your kid could be doing without all that commute time. Or maybe you want to move to San Jose? Good luck either way you have quality choices!

When making this choice, don't forget to consider "happiness quotient" which often revolves around student electives, extracurriculars and friendships. Does your student like art? Music? Sports? Taking commute time into consideration, in which school would they be able to pursue and enjoy such interests? Which school might provide the best potential friend connections? I suspect that ten years from now, you'll look back and feel this decision with regard to academics was less important than you think it is now- that a fine education was available at each school; socio-emotional well-being might be the more important consideration.

@John Jacobs,
That's a good point, and there is another problem with this whole region, especially PAUSD, which is that before the date for starting school was moved back, teachers were regularly telling parents to hold baxk their children before starting school. This led to a huge range in ages in the same grade. In middle school, that difference becomes exacerbated as children go through puberty. Imagine being a late-blooming boy as boys two years older than you in the same grade set the standard. The district never addressed this problem, it was the parents' fault for sending their kids to school when the law said they should (never the district's fault for making young fives a choice rather than a needs based or teacher referral program).

Middle school is difficult enough socially. If I were choosing now, I would put that issue on the radar. Hopefully by now it is worked out at the middle school level but I don't remember when the law changed.