Is there a separate app for Seeking Alpha PREMIUM that I need to download?

No.

The Seeking Alpha app for iOS devices already includes your PREMIUM functionality and features. You simply need to log in with your PREMIUM account.

Once you are subscribed to PREMIUM, you will automatically have access to our archive of over 880,000 articles and transcripts covering more than 10,000 stocks.

You will notice some articles have a key symbol next to the title, these are the articles PREMIUM gives you access to.