I do not know what kind of photography you mainly do, so it is difficult to recommend. I do not live in Stockholm (or Copenhagen), but I have visited Stockholm quite many times. I have not managed to get a decent photograph there, though, but that is my problem.
There are of course the standard tourist attractions, the old town and the castle, if that is what you are interested in. There is a nice photography museum http://fotografiska.eu/
in Södermalm, and strolling around Södermalm is also fun. The "Vasa" ship is a must-see, of course, the high point of Swedish engineering (or more accurately, a monument to excessive customer requirements). A ship that sank on it's maiden voyage from the first breeze it encountered in 1628. You'll need a wide aperture lense when photographing the ship, at least the last time I saw it the lights were dim to shield the old ship. Or boost your ISO, as you probably need some depth of field as well. Wide angle is probably ideal.
If you are into concerts, there are quite many of those in June: Zara Larsson June 1st, Veronica diMaggio June 2nd, Radiohead June 9th, and Justin Bieber June 10th. And lots more, they may be sold out, though.
Stockholm is situated on islands, there is plenty of sea to see, and boat trips to make. If you are from the States, you probably like driving a car. You could make a scenic drive (well, not all of it is scenic, but quite some of it is) around the islands, which also involves two very short ferry trips; from Stockholm to Värmdö (road) to Myttinge to Rindö (ferry) to Vaxholm (ferry) and back to Stockholm. Check the schedules, though.
And if you want to see the real multi-cultural Stockholm, you could take the subway to e.g. Kista (a suburb) during daytime, when Kista is a business city, and when the evening comes the population changes totally. But well, if you are not into that kind of thing, maybe not. Watching all the different nationalities at Sergels Torg may be enough.
And as mentioned the light will be great. You'll be able to photograph practically through the night, although the sun does set at Stockholm's latitude for a few hours. But it won't be pitch dark even in the middle of the night. Sunsets are late and sunrises very, very early. (Quite many foreigners find it difficult to sleep, actually.)