What is the difference between Watson Assistant (formerly Watson Conversation) and Watson Assistant Solutions

I'm only familiar with Watson Assistant because I've done some work with it but I've never used Watson Assistant Solutions.

Is it a new API or an app on top of Watson Assistant? When should I use one or the other?

They are both IBM Cloud/SaaS services. You can learn more about them here:

Watson Assistant (formerly Conversation Service)

• Build multi-turn natural language dialog.
• Catalog of already configured customer service and skill workspaces.
• Insights into conversations and improve your training
• Built-in system entities to ease conversation design
• Doc, Boilerplate apps, REST API, SDK and skill Workspaces
• Publish AI bot channels for Watson Assistant AI bot to Slack, Facebook, Messenger, or Twilio.

Watson Assistant Solutions An AI assistant framework, services and starters that you use to build your own branded, personal and proactive multi modal personal assistant. Optimized for use with any IOT device.

Watson Assistant for Industry

• Out of the box skills: Weather, Greeting, Conversational Essentials, General Knowledge and IFTTT. Includes Watson Assistant (Formerly Watson Conversation Service) workspaces.
• Device enabled for allowing users to communicate via Audio Gateway for Voice
• Contextual skill routing service using via intent, location, language and previous skill context
• Doc, SDK and Skill Boilerplate to build new custom skills

Watson Assistant for Hospitality

• Includes Watson Assistant for Industry
• Out of the box hospitality skills for Service Requests, Venue Information, Reservations, Command and control of devices & more

Watson Assistant for Automotive

• Includes Watson Assistant for Industry
• Out of the box automotive skills: IOT for Automotive to understand car data, Points of Interest, Navigation and Car Manual
• Business Console for administrating skills