The main differences between InnoDB and MyISAM ("with respect to designing a table or database" you asked about) are support for "referential integrity" and "transactions".
If you need the database to enforce foreign key constraints, or you need the database to support transactions (i.e. changes made by two or more DML operations handled as single unit of work, with all of the changes either applied, or all the changes reverted) then you would choose the InnoDB engine, since these features are absent from the MyISAM engine.
Those are the two biggest differences. Another big difference is concurrency. With MyISAM, a DML statement will obtain an exclusive lock on the table, and while that lock is held, no other session can perform a SELECT or a DML operation on the table.
Those two specific engines you asked about (InnoDB and MyISAM) have different design goals. MySQL also has other storage engines, with their own design goals.
So, in choosing between InnoDB and MyISAM, the first step is in determining if you need the features provided by InnoDB. If not, then MyISAM is up for consideration.
A more detailed discussion of differences is rather impractical (in this forum) absent a more detailed discussion of the problem space... how the application will use the database, how many tables, size of the tables, the transaction load, volumes of select, insert, updates, concurrency requirements, replication features, etc.
The logical design of the database should be centered around data analysis and user requirements; the choice to use a relational database would come later, and even later would the choice of MySQL as a relational database management system, and then the selection of a storage engine for each table.
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