First of all a Integrated Analytics term is used with reference to a software or platform which has a analytical capability tool. For instance , if its a simple CRM tool, ie Customer Relationship Management tool like zoho, Salesforce, SAP , Oracle CRM, Microsoft Dynamics, Nimble, Sugar, Hub spot and so on, they might list down their tools features , one of which would be an Integrated Predictive tool. To summarize it means , all the data which is kept in the system can be harvested using its integrated analytical tool to create insights using charts/dashboards/etc. This Integrated tool could be an engine which might be using visualization of data pipelines or go ahead and use models to show diagnostic as well as predictive reports. Some of these reports and insights might be default basis the application and some could be custom designed. For instance in most Sale's CRM, Lead conversion, lead life would be displayed as a analytical cube within the datawarehouse. Additionally grading of leads like high probable/Low Probable could be an output of the custom analytical functionality, which would take some time post substantial usage of platform and learning by the tool.
A Unified Analytics Platform on the other hand talks about a set of analytical tools bundled together for delivery of Analytics. This could be a cloud or on premise deployment depending on the vendor format. You can also think of it as a suite of all tools required to ingest, stage, extract, transform , load, process, model , visualize, predict, test and deploy data model. Typically if you think about systems like SAP, which have database, data warehouse, data analytics, reporting - BI/BO, modelling - PAL (Predictive Analytics Library, HANA - Column based data, and so . Its a suite which has all the tools required , can be purchased together or in pieces. There are others like SAS , Oracle etc which have similar infrastructure. On the Cloud front platforms like AWS (Amazon Web Services), Microsoft (Microsoft Azure) and GCC (Google Cloud Console) can be classified as Unified Analytics Platform.
One of the best example of Integrated Analytics tools is Google itself. So if you have ever had a website of yours and wanted to analyse the data with respect to source of users, time of day the traffic is landing on your page, how long do they stay on your page, bounce rate of your traffic , live users count , and so on . There are more than 100 diagnostics you can check and it keeps on growing thus helping you to understand what is wrong with your page or what is right with your page. So be it social or business needs, the integrated Google Analytics tool helps you to optimize your page. This i would say is the best example of Integrated Analytics tool.