In a stacked laser diode array, multiple beams are emitted parallel to each other. These beams can be very close to each other, which should results in interference patterns. How does this setup differ from the scenario when light travels through an aperture with multiple slits, producing a large diffraction pattern behind it (as in the picture below)?
I see three possible reasons to explain why you do not see the interference pattern shown in your picture.
1) The laser diodes are not coupled and hence there is no reason why the laser diodes should be in phase. The pattern in your picture is created if the slits are illuminated by a plane wave which means that the "light sources" represented by the slits are in phase. Any phase offset between the light sources will create a different interference pattern but will still differ from the interference pattern of a single slit. The single wavefront equivalent would be an illumination of the multiple slits with a warped wavefront.
2) Laser diodes are not stabilized and are therefore prone to mode hops (which are accompanied with phase changes) and wavelength shifts in combination with phase shifts. This temporal phase instability between the different light sources creates varying interference patterns (see reason 1) which might change faster than the eye can see. An example of fast mode hopping is e.g. given here on page three. The single slit interference is not influenced by the phase shifts (the wavefont is still be assumed as constant within the width of the slit). Therefore, in the extreme of fast random phase shifts, the observer will see the interference pattern of a single slit only.
3) Only light with identical wavelengths can create temporally stable interference. A set of laser diodes have usually slightly shifted spectra. Therefore, only the overlapping spectrum of the different light sources produces a multi-slit equivalent interference (and single split interference) and the non-overlapping spectrum single-slit interference only.
The difference is that unless the laser designer took special care to make it so, the output of the lasers in the array will not be mutually coherent. (Also, doing so would generally require making the laser array on a single chip, rather than stacking multiple chips as described in your linked document)
Therefore the interference pattern they produce will be constantly changing, on a time scale too fast for you to observe the interference by eye.
In the case described in the document you linked (multiple laser devices used together to produce high power), it wouldn't be desirable to have mutually coherent output (precisely because it would produce an interference pattern) so they certainly haven't gone to any special trouble to make them coherent.