Setting aside hillside lots, lets consider a relatively flat piece of property for a moment, when evaluating building a home addition, either going up (adding a second story) or out (staying on one level).
Building a one story home addition onto the rear of the home will take up yard space, thus limiting outdoor play/recreational use, as well as limiting landscaping, pool and gardening abilities. Also, the longitudinal axis of the home will be projected, thus more of a distance to get to the entry door, front yard and parked vehicles. On the other hand, natural light via skylights as well as volume ceilings can be installed though most of the home, creating very pleasant and energy efficient spaces. Also, a one story set of plans can be approved of much faster by the building department and the addition can be built faster.
Building a second story home addition will provide a more ‘compact’ home, without taking up much yard space. Traversing up and down stairs is a consideration, that should be viewed as a long term condition. Logistics of access to the kitchen, garage and outdoor areas, is likely to be a less frequented decision, but will also likely be thought out much better. Architectural plans will take longer to approve of by the Building Department and permit fees a little more costly. Enjoying views that may be available is a big plus, along with first / second floor privacy that can be enhanced.
The structural part of the building will be an important influence to cost. The footings for a one story home addition, given acceptable soil conditions, can be of modest depth, thus less digging and less concrete needed. Also, the framing elements for a one story addition can be of all timber members. The other trades have less of an impact on cost.