The emergence of tools that support fast-and-easy visualization creation by non-experts has made the benefits of InfoVis widely accessible. Key features of these tools include attribute-level operations, automated mappings, and visualization templates. However, these features shield people from lower-level visualization design steps, such as the specific mapping of data points to visuals. In contrast, recent research promotes constructive visualization where individual data units and visuals are directly manipulated. We present a qualitative study comparing people's visualization processes using two visualization tools: one promoting a top-down approach to visualization construction (Tableau Desktop) and one implementing a bottom-up constructive visualization approach (iVoLVER). Our results show how the two approaches influence: 1) the visualization process, 2) decisions on the visualization design, 3) the feeling of control and authorship, and 4) the willingness to explore alternative designs. We discuss the complex trade-offs between the two approaches and outline considerations for designing better visualization tools.