The choice to use online or desktop is a purely personal one. Online you get to file one return with efile per account. Your return is saved online. The desktop you can file up to 5 returns with the same program. Your return is saved on your computer. For online the state return is an additional fee of $59 with efile. For desktop 1 state download is included but there is a $25 fee to efile.
Premier desktop includes a download of the state software of a single state. With that state software you can prepare multiple state returns of that same state, but there is a $25 state efile fee (except no efile fee for NY state.) Of course, there is no efile fee if you choose to print and mail the state return. Federal efiles (up to 5 Federal returns) are free.
The desktop version has some advantages over online Premium. The desktop version has Forms Mode for easy viewing (and even editing) of forms during return preparation. Printing can be easier. And the ability to prepare multiple returns as mentioned above.
With the desktop edition, all of the tax data files and PDFs are stored locally on your computer. So you have to be careful to back them up in case of a computer crash, and store them safely and securely (and guard against theft), so you can use the tax data file the next year to transfer into a new return.