As everybody described Lithuanian part of Curonian Spit I will focus on Russian part of this cultural heritage site. Visited in July, 2019, while it was possible to visit Kaliningrad Oblast (Królewiec Area) without Russian visa.
Access from Polish side - border crossing at Grzechotki / Mamonovo - 2 by own car, and then normal transit via Zelenogradsk towards Curonian Spit National Park (Национальный Парк Куршская Коса). Entrance to the park from Russian side is paid - 150 rubles per person and 300 rubles per vehicle. The point where they collect money is located here 54°58'33.0"N 20°31'13.7"E (Kontrol'no-Propusknoy Punkt Natsional'nogo Parka Kurshskaya Kosa, контрольно-пропускной пункт национального парка Куршская коса). The booth is located some 30 meters from the main road, so if you want to buy the ticket to enter the park, you just stop the car on the road (there is no parking space) blocking the passage. There is always a huge traffic jam before the ticket booth in the morning (the queue usually starts at Zelenogradsk - 2-3 km from the booth).
Russian part is pretty similar to Lithuanian one but for me the villages are far less preserved. First on a way, Lesnoy looks like any other Russian village - many new houses built of white bricks. Rybaczij (Rossiten) is a little bit more similar to German towns (some old buildings are still preserved) but overall impression is somehow disappointing. As far as I remember Lithuanian villages are far more better preserved and maintained.
Interesting points on Russian part of Curonian Spit are certainly: Dancing (or Drunk) Forest (Танцующий лес, free entrance) – mystery pine forest where trees are bent to different (natural phenemon of unknown origin – there are different hypothesis, no one is fully proved), Fringilla Bird Observation Station (Биологическая станция "Рыбачий" Зоологического института РАН, Полевой стационар "Фрингилла", as far as I remember entrance fee was around 300 roubles per person, half an hour guiding tour throughout the station - you will see how they catch and ring the birds) and numerous sand dunes located on both sides of the road. In July, 2019, area around Lebed Ozero (Swan Lake) was closed due to conservation reasons.
Practilities: I stayed overnight in Rybaczij at Усадьба "В камышах" (officially translated as Manor "The reeds", 55°09'07.3"N 20°51'27.2"E), nice place, modern building but with some style. In the village there are several shops (on the main street - ulitsa Pobedy) as well as some restaurants.
Photos: top left, then clockwise, Sand dunes, Fringilla Bird Station, Dancing Forest, Old School at Rossiten