In 1872 the Grand Orient of Belgium removed every reference to a Grand Architect of the Universe from its statutes. More famously, the Grand Orient of France followed a decade later. Both Grand Lodges lost recognition from the United Grand Lodge of England and thus became “irregular”.
Both remained -at the time- men-only organisations until 2010 (GOdF) and 2021 (GOB). By that time mixed gender Freemasonry had been formed (1893 Le Droit Humain) and women-only Freemasonry (1935 Grande Loge Féminine de France). Le Droit Humain quickly spread across the globe and gave birth to split-off Grand Lodges. Also women-only Freemasonry started to spread to other countries. Also the example of the GOB and GOdF was followed by more men-only organisations and thus a very varied ‘other form’ of Freemasonry developed.
A large part of French Freemasonry has “laïcité” (secularism) high on the agenda, but this is not an essential element of “Continental Freemasonry”. There are more than a few Grand Lodges who do have the requirement a belief, but which are still “irregular” to (for example) the United Grand Lodge because they admit (only) women (again but an example).
What you do see in “Continental Freemasonry” is more variety within organisations. It is not uncommon to have a Grand Lodge, Grand Orient, confederation or whatever form of organisation, which has lodges which are secular and lodges which are not. And/or lodges that admit (only) women and lodges which do not. This variety of lodges within one organisation is not possible within the boundaries of “regular” Freemasonry.
As you can see a patchwork of Freemasonry is formed outside the scope of the ‘traditional’ Grand Lodges. It is not easy to catch all this variety in a single definition. Some speak of “adogmatic”, others of “modern” Freemasonry. Those who see the Grand Orient de France as the centre of this ‘other branch’ use “Continental” referring to the continent of Europe. That is not to say that a continent such as South America has no such forms of Freemasonry. Quite the contrary!
Since the term “Continental’ seems to gain some recognition (it’s the main lemmet on Wikipedia), it is used as title for this website, imperfect as it may be.
Last but not least, it is not like all “Continental” organisation recognise each other. Things are not so clear-cut that there are two camps within Freemasonry (“regular” and “irregular”). In neither camp there is an overarching super Grand Lodge or an organisation that dictates how things go. So you can have a Grand Lodge that is recognised by one “regular” Grand Lodge, but not by another. Similarly there are “Continental” organisations that recognise some other organisation while others do not.