Transifex Native is a full suite for cloud-based localization for a continuous development flow, consisting of many parts (SDK, fileless communication, Content Delivery Service-CDS). The part that mostly resembles the API and the Transifex Client is the Transifex Native SDK so we will focus on that.
Compared to the API and Transifex Client, Transifex Native SDK handles most of the communication with the localization services automatically from day one without the need for manual actions by the developer. Were you needed several API calls to push your localization file to Transifex or get translation files back, with Transifex Native you need a single push command that will detect and push all found content to the CDS while translations are automatically pulled on request.
Additionally, there is no need for management of your localization files and content in your working environment (localhost) like you used to do with Transifex Client. The only similarity currently shared with Transifex Client is the push command that will send detect content to a localization service, although the specifics are quite different due to Transifex Native handling phrases while Transifex Client is handling localization files.