Part of what makes superheroes who they are and attracts readers/viewers to them is their origin stories. Superman is an alien with super powers because of his alien status combined with the light of a yellow sun. He's not "one of us", he's an external savior.
One of the central concepts of Shazam is that he could have been anyone. In the film he's Billy, a 14-year-old boy who gets the powers from an old wizard after being chosen:
A boy who is chosen as the "Champion of Eternity" and given powers by an ancient wizard, and transforms into an adult with superpowers when he speaks the sorcerer's name.
This means that anyone could be Billy. Anyone has the chance to become the "Champion of Eternity". It's not decided by something inborn like what your species is, but by the person inside. While you can't ever be Superman because you're human, you could be Shazam. This gives a slightly different appeal/audience to the character.
On top of that, the fact of him really being a teenager gives filmmakers more time to have fun with the character, which is evident in the trailer. You get a comedic superhero film instead of a serious one - I think we can all agree that the recent Superman films are a bit... grey.
So, that's the main answer for me. Different origin story and different tone make a different character even if the powers are somewhat similar...
For reference, Shazam's full slate of powers are somewhat broader and slightly different than Superman's:
These powers include: the wisdom of Solomon, granting increased intellect; the superhuman strength of Hercules, the superhuman stamina of Atlas; powers from Zeus, including the ability to summon lightning as well as being nearly indestructible; the increased courage of Achilles; and the superhuman speed of Mercury, which provides him with the means of flight.
For example, control of lightning isn't really quite the same as shooting lasers out of your eyes.
As to whether it's part of the same DC universe, one of the longer previews (second in the "trailer" link above) shows icons of Batman and headlines in newspapers about Superman (1:34 in timecode) but the film is set in Philadelphia (according to the Wikipedia article) and Billy's foster brother is referred to knowing things about "this caped crusader stuff". The "Certificate of Authenticity" in the image below from the preview may indicate that everything in the drawer is an artefact from a superhero film/TV show rather than being from the world they live in.