Generally it's easier to book a trip with legs on different carriers if they have a codeshare arrangement, because you can just do it as a single itinerary through one carrier's web site. Otherwise you often need a travel agent to be able to put together the itinerary (that's not strictly necessary though, just the way they make it work so no technical reason they couldn't just offer non codeshare flight numbers)
The other thing is airline points/status - sometimes you get no status points/miles on non-codeshare partner flight numbers but you do on the codeshare flight numbers.
The other thing is airline points/status - sometimes you get no status points/miles on non-codeshare partner flight numbers but you do on the codeshare flight numbers.