Power BI allows to add calculated field and measure to table. Both create new column and allow me to add DAX formula.
When to use calculated field vs measure in Power BI?
The most important difference is that calculated columns are calculated once when the dataset is loaded. Their value does not change later, i.e. it is not affected by slicers for example. Measures are dynamic. They are calculated whenever necessary, thus they will respond to slicers in the report.
I would recommend to read this article - Measure vs Calculated Column: The Mysterious Question? Not!
Regarding similarities between calculated columns and measures, both are:
Calculations that you can add to your data model.
Defined by using a DAX formula.
Referenced in DAX formulas by enclosing their names within square brackets.
The areas where calculated columns and measures differ include:
Purpose - Calculated columns extend a table with a new column, while measures define how to summarize model data.
Evaluation - Calculated columns are evaluated by using row context at data refresh time, while measures are evaluated by using filter context at query time.
Storage - Calculated columns (in Import storage mode tables) store a value for each row in the table, but a measure never stores values in the model.
Visual use - Calculated columns (like any column) can be used to filter, group, or summarize (as an implicit measure), whereas measures are designed to summarize.