By Joe Flint and Lillian Rizzo

HBO Max, AT&T Inc.'s new streaming service, launched Wednesday but remains unavailable to Amazon.com Inc. Prime Video subscribers, as both companies remain at loggerheads over where the content will reside and who will have access to user data.

Amazon has about 5 million customers who get HBO through its Prime Video Channels platform, all of whom aren't able to access HBO Max, which is being offered to existing HBO customers at no extra charge. Amazon subscribers who want the new version would have to drop HBO from their current service and subscribe directly to the platform, which costs $14.99 a month.

"AT&T is choosing to deny these loyal HBO customers access to the expanded catalog," Amazon said in a statement. "We believe that if you're paying for HBO, you're entitled to the new programming through the method you're already using. That's just good customer service and that's a priority for us."

A spokesman for AT&T's WarnerMedia unit said Wednesday that the company was looking forward "to reaching agreements with the few outstanding distribution partners left, including with Amazon and on par with how they provide customers access to Netflix, Disney+ and Hulu on Fire devices."

HBO Max also has yet to reach a deal with Roku Inc., and wasn't available to cable giant Comcast Corp.'s existing customers when it launched. Comcast and WarnerMedia reached an agreement Wednesday morning to give Xfinity cable customers access to HBO Max.

Clashes between streaming services are becoming increasingly common, and in many ways are akin to the fights that traditional cable networks often have with cable and satellite operators. Walt Disney Co. and Amazon disagreed over terms regarding Disney+ only to reach an accord in time for that service's launch last November.

At the center of the dispute between Amazon and WarnerMedia is that WarnerMedia wants Amazon to shuttle subscribers to the HBO Max platform as it does with Netflix Inc. and Disney+ customers. Amazon, however, wants to house the HBO Max content on Prime Videos Channels as it currently does with HBO.

Essentially, AT&T wants Amazon to serve as a toll road to HBO Max instead of providing housing for it. Such an arrangement would make it easier for AT&T to track consumption habits and other valuable data about its customers and establish a direct relationship.

The row with Roku, a connected-television platform that serves as a gateway to streaming services, is different. People familiar with the matter said Roku was still in a standstill with HBO Max over the terms of a carriage agreement.

Typically, Roku takes a cut of a service's subscription fees and gets to sell ads in return for distribution. There are disputes on both those fronts with HBO Max, which next year will introduce an ad-supported version of the platform.

The standoff in negotiations between Comcast and HBO Max -- which was resolved Wednesday morning -- centered around increased pricing for the streaming version of the premium channel, according to people familiar with the matter.

Comcast, the nation's largest cable operator, with nearly 21 million video customers, was once one of HBO's biggest distributors with roughly 50% penetration, a person familiar with the matter said. Now, that number has dropped to less than 30% of Comcast customers that subscribe to the service, the person added.

For Comcast, which is launching its own streaming service, Peacock, in July, HBO Max now poses more of a threat as a competitor than just an add-on to the cable bundle.

In some cases, distributors have been able to leverage concessions from AT&T in return for agreeing to offer HBO Max. Charter Communications Inc. finally landed a carriage deal on AT&T's DirecTV for the Los Angeles Dodgers channel it distributes for the team in return for an HBO Max pact, people familiar with the situation said.

Other providers who have signed on to carry HBO Max include Altice USA Inc., Cox Communications Inc. and Verizon's Fios TV.

Write to Joe Flint at joe.flint@wsj.com and Lillian Rizzo at Lillian.Rizzo@wsj.com