Are the words 保育所 and 幼稚園 synonyms or is there some distinction between them, either in meaning or formality?
While there's definitely some significant overlap in the services they provide, the two are considered distinct types of facility, overseen by different governmental branches.
A 幼稚園 is a primarily educational institution, officially a type of school, and so they require their staff to have teaching qualifications, only accept children from ages 3 and up, and often have relatively short hours (traditionally from around 9 AM to 2 PM, though many institutions will offer the service of taking care of the children for longer).
A 保育園 on the other hand is a childcare institution, whose primary purpose is to look after children while their parents are unable to do so. As such they have longer hours (from around 7 AM to 6 PM) and no definite lower age limit on the children they will accept. They're also required to provide lunches, which isn't the case for 幼稚園.
Despite the differing focus, both types of institution will of course provide some measure of both educational and care services, and both cater to children up to around the age of 6, when they will enter the compulsory education system by enrolling in a 小学校.