The after being dubbed "too mature" for the former.
For this reason, many people have long seen HBO Max as a potential foil to Disney+. With access to the many, many brand owned by WarnerMedia, along with original HBO programming, the service won't be lacking for kid-friendly material like the newly-released Looney Tunes shorts and the Adventure Time specials. At the same time, their larger animation catalogue is noticeably diverse when it comes to not only style, but also tone and subject matter.
In addition to Looney Tunes, HBO Max either already has or will eventually include lots of anime series from Crunchyroll (Death Note, Berserk, Rurouni Kenshin), The Boondocks, on top of streaming Cartoon Network titles intended for a variety of age groups. And that's not to mention the original animated programming they've got in the pipeline, ranging from Gremlins: Secrets of the Mogwai to the Seth Rogen and Sarah Silverman-led adult cartoon comedy series Santa Inc.
By comparison, Disney+'s animated catalogue includes the Mouse House's many animated features (including, more recent films like Big Hero 6 and Gargoyles, which are known for appealing to a wide range of viewers, most of these titles are geared pretty directly towards kids (with The Simpsons again being the main exception to the rule). HBO Max, on the other hands, includes a whole lot of material oriented towards young adults and/or mature demographics, as well as a more international audience in general.
Far from being a problem, this means Disney+ and HBO Max can compliment (rather than compete with) one another, as far as their animated offerings go. Depending on their interests, customers can thusly pick and choose HBO Max is arguably better than Disney+ (and vice versa), this is one aspect where the difference is more a matter of preference, not quality.