I went to a private Christian high school in Canada, and today I teach at one. The school I teach at is small: around 400 students, compared to 1,000 – 2,000 in nearby schools in the region.
To directly answer your question about religion: This school does aim to register families who subscribe to a statement of faith and more general identity statement. They've also been given welcome sessions to orient them as to what the school is about. The goal of this is mostly to make sure they know what kind of a school they'e sending their children to, so that we can be on the same page as to goals and possible conflicts.
In the past and probably still now, the school has accepted non-Christian families up to a certain proportion of the population. There are certainly students from religious and non-religious families who feel comfortable saying they're not Christians. We also have a course on world religions that's very much about listening and respecting and involves a visit to another faith's place of worship.
Now, to fill in the picture for your decision a bit more...
Moral teachings. Sure, you can count this. Of course, public schools don't discount or omit moral teachings, but I don't know if there's a similar concerted effort to make it a cornerstone of the education. When I teach human geography, I don't just teach the facts of immigration; I teach the disadvantages many immigrants face and prod the students to consider their responsibility towards their fellow human beings, rooting that responsibility in Jesus' moral teachings like "Take care of the least of these brothers of mine."
Also, every Grade 9 student goes on a service trip for a couple of days to serve in shelters/food banks and do street walks for the homeless. In Grade 11 they do the same for longer and with more independence, and also do a simulation of a refugee/migrant camp. On all occasions we give them perspective on these issues and what causes them. Without these our school wouldn't be what it is.
Staff cohesion. The above point is one all my colleagues are together on, because we all meet several times a week before school and we share a devotion, news, prayer, vision for the school,* and so on. There is also less turnover here than at nearby public schools; several people have been here for 15–30 years, whereas the norm in the public system is to be with one board that long but not one school. That means there's a strong core group who understand what we're doing here.
* Note that we're not all cohesive on particular points of dogma, nor do we require any particular dogma from students.
Belonging. One weekly meeting is focused on one grade (rotating each week); all those who teach Grade X get together to share concerns and joys about specific students, from academics to social integration. It's very hard for students to fall through the cracks this way, and very easy for strengths to be identified and cultivated.
Similarly, there are weekly assemblies that are often about enfolding. Recently, one student lost a parent and (with consent) this was shared at assembly and some advice was given for how to support someone in grief. This is possible only because our school is so small.
Room to breathe. In an independent school there's less red tape, especially around certification. For example, I took some computer science at university, did some development work, and TA'd introductory undergrad courses for years. At a public school, that has no bearing on whether I can officially teach it at the high school level, but an independent school is able to give me the course based on a standard hiring process and judge my ability based on the outcome.
Also, there's more room to do unconventional things; for example, we have a "Focus Day" program where about once a month, the students spend all day in just one of their periods. This allows the teacher to do a big hands-on project or a field trip more easily. We also get to do interesting locally developed classes, such as a double-period environment/civics one we have that teaches social responsibility plus sustainability and links the two.
More frequent inspection. In our province, public schools are held to the standards by the school board. They are rarely directly inspected by the ministry because the ministry devolves this responsibility to superintendents. But independent schools are inspected every two years directly by a representative of the ministry. Ironically, this means we're probably held to ministry requirements more strictly.
Incidentally, this also means we offer the ministry curriculum, because we still want our students to get a high school diploma. I don't know if other schools forgo that or what.
Cost. Although is a significant con, it does have one indirect benefit: the parents are heavily invested in the school and often push to get their money's worth, which generally means higher standards for your kids, too.
Politics. Officially, a school's politics aren't visible to the students and don't influence them. In reality, a small independent school's culture is pretty clearly defined. At a Christian school you're likely to have conservative, right-leaning families and the students bring their politics from home and reinforce each other. I leave it to your imagination to fill in the picture of our demographics. In some private schools, there's also a danger of the politics of elitism (though I don't notice it here).
Feedback loop. I work with some teachers who taught me or my relatives and I teach some children of my former teachers. As a school we hire people we know are good, because we don't have restrictions about hiring somewhat randomly from board-wide supply lists as the public schools do. Between that factor and the teachers who've been here for decades — the low turnover I praised above — we have fewer points of contact / less "surface area" with the outside world. Many of the teachers also have their undergrad and/or teaching degree from a Christian institution. While these are certified by the public ministry and we're due-paying members of the public college of teachers, it's not as easy for changes in curriculum or pedagogy to filter in.
We also have less in the way of central professional development; public boards have the resources to hire experts and send them around to all the schools, but we have only a few institutions that serve our type of school. Thus we and our administrators have to do more intentional research to keep up with the requirements of the inspections I mentioned.
Opportunity. This is one that differs widely based on the type of private school. A larger, more elite private school might offer more opportunity and specialized programs like IB. We can't always do that. We don't have enough students to offer languages besides French, nor consistently run standard vs. immersion French classes. Having fewer students also limits some range of competition — both practically (schools with 2,000 students can field teams with more talent, so you compete in lower brackets) and technically (there are contests my students have been denied access to because they're not in the public system). Of course, this is traded off with the "room to breathe" above: there are things we do that other schools don't.