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In my limited experience, all game-viewing is a matter of luck. You will almost certainly not see everything you hoped to see, wherever you go: the wild is wild, and many animals are doing their best not to be seen. So no place can guarantee you will see what you want to see, but you can have a fantastic experience anyway. Suggestion: buy a birding guide. Kruger has some absolutely beautiful birds (I'm sure Serengeti does too), and many an hour of looking in vain for lion or zebra has been rescued by spotting some of them.
I have visited Kruger National Park (KNP) in June or July, but never the Serengeti. This answer will therefore be based on a mix of experience and online info.
June is the cool-and-dry season in both KNP (average of 11mm of rain; 6-26Β°C) and Serengeti (31 mm; 15-26Β°C). This means the vegetation is not so attractive, but makes it easier to spot game as there is less foliage in the way, and animals must gather at watering holes to drink.
In Kruger National Park your budget should easily cover anything you want to do (the best accommodation in the park, plenty of guided walks and drives, and eating in restaurants). I don't have much experience of what the higher-end accommodation is like, but it's probably not five-star. However there are private game reserves nearby which offer much more expensive experiences. You may wish to visit both (it's also common to stay in a lodge in a private game reserve, and visit KNP on some days).
Perhaps the best-known attraction of the Serengeti is the Great Migration of wildebeest and other grazers. According to https://www.expertafrica.com/tanzania/info/serengeti-wildebeest-migration this pauses in June; the most spectacular time is in July.
One big difference, apparently is that KNP is accessible: it's a relatively short drive from Johannesburg, and South African schools will be on holiday until the 24th of June. This means the park may be quite busy in places (the southern part much more than the north).
But Kruger is also more accessible in a different sense: if you rent a car, you can tour Kruger pretty much independently. While I recommend taking guided drives and walks as well, you can just take your car and drive in any direction, move randomly across the park, and end the day at the same or another camp. (You must return to camp between dusk and dawn, though, unless you're on a guided drive). Apparently Serengeti is much less accessible in this way.