How similar can an app be to another one before there are legal problems?

Depends on what is being "protected" (see current trends in trade-dress and UI/UX design elements). Copyright avoids outright cloning, trademark passing off and patents novel advances in the state of the art. So if the cloners are not infringing, there is no legal basis for recourse. This shows that a first mover advantage is only temporary which is what you'd expect in a competitive market so it is up to the app-owner to keep on innovating and building value, usually around some variant of a trade secret.

The owner of the prestigious app can potentially seek redress using actions under patent, trademark, and/or copyright law. If the similar app performs these services using a method that is protected by a patent held by the owner of the prestigious app, then the patent owner may be able bring a patent infringement suit against the owner of the similar app. If the mark (e.g., name/logo) used on the similar app is confusingly similar to a mark associated with the prestigious app, the owner of the prestigious app may be able to bring an trademark infringement suit against the owner of the similar app. If the graphical screens (e.g., images, text, etc.) presented by the similar app are similar to those presented by the prestigious app, the owner of the prestigious app may be able to bring a copyright infringement suit against the owner of the similar app.
-The above is not legal advice -