Spotify claims that Apple's App Store rules deliberately "limit choice and stifle innovation at the expense of the user experience," partly due to the 30% cut Apple takes from purchases made through its system, and partly because Apple is "both a player and referee."
The streaming service's CEO Daniel Ek wrote, "If we pay this [30%] tax, it would force us to artificially inflate the price of our Premium membership well above the price of Apple Music." His statement continued, "If we choose not to use Apple’s payment system, forgoing the charge, Apple then applies a series of technical and experience-limiting restrictions... Apple also routinely blocks our experience-enhancing upgrades" which includes access to Apple services such as Siri, HomePod, and Apple Watch.
Apple has since responded with a statement of its own. The statement is quite lengthy, but essentially Apple refutes the claims, stating, "After using the App Store for years to dramatically grow their business, Spotify seeks to keep all the benefits of the App Store ecosystem... without making any contributions to that marketplace. At the same time, they distribute the music you love while making ever-smaller contributions to the artists, musicians and songwriters who create it — even going so far as to take these creators to court... Just this week, Spotify sued music creators after a decision by the US Copyright Royalty Board required Spotify to increase its royalty payments."
Apple refers to Spotify and other companies including Google, Amazon, and Pandora appealing the US Copyright Royalty Board (CRB) for clarification over increasing royalty rates. It's fairly complicated, but essentially, Spotify isn't suing creators. Also, as The Verge points out, whatever result the CRB hands down will apply to Apple Music, too.