Neural language models have the potential to support human writing. However, questions remain on their integration and influence on writing and output. To address this, we designed and compared two user interfaces for writing with AI on mobile devices, which manipulate levels of initiative and control: 1) Writing with continuously generated text, the AI adds text word-by-word and user steers. 2) Writing with suggestions, the AI suggests phrases and user selects from a list. In a supervised online study (N=18), participants used these prototypes and a baseline without AI. We collected touch interactions, ratings on inspiration and authorship, and interview data. With AI suggestions, people wrote less actively, yet felt they were the author. Continuously generated text reduced this perceived authorship, yet increased editing behavior. In both designs, AI increased text length and was perceived to influence wording. Our findings add new empirical evidence on the impact of UI design decisions on user experience and output with co-creative systems.