On one ticket your bags check through, the airline is responsible for protecting your connection, and your assistance requests carry across the whole trip. On separate tickets you collect bags, re-check them, clear security again at each stop, and rebook at your own cost if a flight runs late.
Booking sites will happily assemble a cheap multi-city trip out of separate tickets on unconnected airlines. It looks like one itinerary on the confirmation page. It is not one itinerary, and the difference shows up at the worst possible moment.
On separate tickets between airlines with no interline agreement, bags are not checked through. At the connecting airport that means clearing immigration, collecting the bags, queueing at a different check-in desk, and clearing security a second time, often many hours into a travel day. If the first flight runs late, the second airline owes you nothing, so you rebook it yourself, at the counter, at your own cost.
One ticket across cooperating carriers gives you bags checked through to the final destination, a connection the airline is responsible for protecting, and one place to go when something breaks.
It also matters for assistance. Requests recorded in the booking, such as the DPNA code, travel with you across the whole itinerary instead of stopping at the first airline.