For example: Day 1 IAH-LAX. Day 2 LAX-ORD.

Could be booked as 2 separate tickets, or as one multi-city reservation.

What (if any) are the pros & cons of each?

(Assume price of either is the same to within a few dollars.)

Does the argument change if the journey is not continuous? - e.g. the second flight might be SAN-ORD instead of LAX-ORD.