what is the difference between turbotax premium and turbotax premier?

Premium is an online software version.    Premier is a version of the desktop download.   The similar names are confusing.   There are significant differences between using online software and using the desktop download.

When you use Online TurboTax, you get one return per fee.  Each return needs its own account and user ID.   If you use the same account and user ID for a second return, you overwrite the first one and lose it forever.

https://turbotax.intuit.com/personal-taxes/online/

 When you use Desktop TurboTax, which you download to your own hard drive, you can prepare and e-file multiple returns for the cost of the software.  You can e-file up to 5 Federal returns—more if you mail the additional ones.  If you have multiple returns to prepare for family members, Desktop is more cost effective.

Desktop --unless you specifically buy federal  Deluxe only from a retailer-- will include the software for one specific state.   You can prepare as many returns for that state as the number of federal returns you prepare.  E-filing state returns is an extra fee which can be avoided by printing and mailing the states.   Additional--other--state software is an additional fee for the software.

https://ttlc.intuit.com/questions/1901306-how-many-times-can-i-e-file-using-the-turbotax-cd-or-download

https://ttlc.intuit.com/questions/2750704-what-are-the-advantages-of-the-turbotax-cd-download-software-over-the-online-version

If you use online software and want a "Live" expert to help then you can choose to use Live Premium.

https://turbotax.intuit.com/personal-taxes/online/live/how-it-works.htm