A friend who ordered a new website recently for her psychotherapy practice told me that the website provide had offered WordPress or Squarespace, and asked which I would recommend. My first thought is that the contractor were 'pulling a fast one' because anyone with a low level of skill can make a Squarespace site. On reflection, the same is true of WordPress, and if the site does the job the client wants, it does not seem improper to use.
Self-hosted WordPress, as distinct from WordPress's hosted service, is just that: you download the software free (here), install it on a server or hosting service of your choice, and do what you like with it. Squarespace is only a hosted service: you can never edit or change the code in most of the underlying software. The most important consequence is to limit the functions you can add to your website. With WordPress you can develop the site in any direction you can imagine, for example adding complex ecommerce features. In many cases, and for large sites, WordPress will not be the best tool, to be sure, and it may require a considerable effort to bend it to your will (although there is a huge amount of ready-made software, not all of it free, for WordPress, which can cover the needs of most smaller organizations). With WordPress you can also take control of the markups and the other bits of the page which end-users do not see but which did, and to some extent still do, influence the rankings you get in search engines, whereas on a SaaS (Software as a Service) service like Squarespace you may get good markup (though this is not true of all Squarespace's competitors) but your control may be limited.
On reflection I may have been unfair about Squarespace. Many clients will only ever want a brochure site, or a photo gallery, and Squarespace seems to do this rather well, and offers many beautiful layouts out of the box. I was influenced by a bad experience on the one occasion when I did try to use Squarespace for a client. He is an artist, and had very limited funds for the website, so I wanted to provide something easy to edit and require minimum developer time to get up and running. It was not a success. Squarespace had just released a new version of their software and it proved buggy, and did not offer some very basic functionality which the client wanted, so we switched to WordPress. Of course I did not charge the client for time I wasted on my 'error' of suggesting Squarespace. I am sure it is better now. If the website contractor is providing what the client wants at a price the client is happy with, and if Squarespace helps to reach that goal, there is no reason not to use it. A website built on SaaS where access to the underlying software is out of your hands had the undoubted advantage that the website owner or developer can and must leave dealing with bugs and keeping out hackers to the company who run the platform. Provided they do that job well, this is a great advantage. I have seen all too many self-hosted sites which have been hacked, require expensive repair and security hardening.
It is only fair to mention that there are several services similar in kind with Squarespace. One of the longest-established and best known is Wix, who seem to advertise heavily on YouTube.
Ultimately for most use cases we at DigitProfessionals are likely to recommend software over which the developer has complete control, such as WordPress or Drupal, or even sites built on fully custom software for very large projects, because we work mainly on hand-made sites: although template can look very lovely, and is right where the budget is small, a site which is individually made usually does a far better job, often at a subtle level, of communicating what the website owner is about. To be sure, a lot of the visual control is available with tools like Squarespace or Wix, and yet the day may come later and perhaps sooner, when we or the client want some custom detailing which bumps up against the limitations of those tools. Anyone who tells you that WordPress or Drupal are similar limited is betraying their own lack of skill at manipulating the software, although it is true that WordPress can be a struggle for very highly customised sites (and a breeze for simple or moderately customised sites), whereas with Squarespace it is entirely foreseeable that the client will want something which is impossible.
Having said all that, for a DIY site, or for a site where fairly modest requirements are clearly within what a tool like Squarespace can do, and are likely to remain so, it may be the best option.