The LotusLive site describes LotusLive Connections as:
LotusLive Connections is an integrated suite of Web collaboration services that lets you store and share files, manage activities, conduct online chats, and extend your business contacts with networking capabilities.
LotusLive Connections is complementary to LotusLive Engage and includes the same exciting set of core collaboration services for customers who want to work outside the boundaries of their company’s e-mail, but do not require Web conferencing or application sharing.
The power of LotusLive Connections is the security-rich, seamless integration between services. It allows you to share and edit information and manage activities easily with one person or a large community of colleagues inside and outside of your company.
As a Web service, you only need a Web browser and Internet connection to take advantage of the exciting LotusLive Connections features and services.
This packaging will work well for individuals or organisations that don’t need web conferencing, or already have access to a competitive service (say Webex) or have an internal on-premise solution available (Lotus Sametime for instance).
I’ve also seen Engage and Connections compared as follows:
What is Engage? Engage is a complete set of integrated collaboration services to simplify and speed up your business projects and interactions. Hold online meetings, share files, track projects, have an instant message chat session, gather feedback and report for your projects and cross-company groups and teams. It’s easy to get your business network together, no matter what company they work for, manage the interactions and share the most accurate information that matter to the project.
What is Connections? Connections is a core set of integrated collaboration services to provide to any active project and company members. The simplified set of services enables any member to post and share any type of file, create and participate in projects and assigned tasks, reach out to anyone in the company or LotusLive network with instant messaging.
Hopefully this helps in distinguishing between the two similar offerings.