An analyst analyzes data. You have a question that needs to be answered with data, so you go to the analyst. They gather data, they look at the data, and they answer your question. Sometimes they find something in the data without your prompting and make a report about it to you. They use straightforward statistical analysis and basic modeling techniques such as linear regression, decision trees, clustering, and exponential forecasting. They use tools with lots of off-the-shelf functionality like SQL, Excel, SAS, QlikView, Tableau, domain-specific modeling tools like Emblem, and occasionally R and Python, where they might have basic proficiency (especially for messy data cleaning). They might be familiar with the fundamentals of probability. They have a keen nose for erroneous data points. They make use of clear data visualizations to carry out their analysis and demonstrate results.
A data scientist carries out research using data. You have a business question or a problem that needs to be solved, so you call a meeting with the data scientist to see if they can help. They determine if and how data can help, and carry out research that addresses your question. They work with you to help define what it is that you need. They have most or all of the abilities of a data analyst. They have a wide variety of tools at their disposal for collecting data, analyzing data, and developing powerful models. They have a deep mathematical understanding of at least a few of those tools, and they know what they don't know about the others. When familiar methods fail, they are able to research new or different methods, which might only be available in technical literature. When existing software fails to implement a technique or algorithm in the way that's needed to solve the problem according to business specification, they are able to implement it themselves in a programming language. They are likewise able to obtain data from a variety of sources and implement data cleaning procedures that might not be available off the shelf. They occasionally work with software engineers to implement their solutions in production systems.