South Sudan has a tropical climate, hot year-round, with a long rainy season from roughly April through November and a dry season from December to March.
Dry season (December–March) is the only realistic time to travel. Roads — such as they are — become passable. Wildlife concentrates around water sources. Temperatures in Juba hover between 20°C and 37°C, hot but manageable. This is also when the antelope migration in the Boma-Bandingilo landscape is most spectacular, peaking around November–December and again in March–April at the turn of seasons.
Rainy season (April–November) transforms the landscape. The Sudd expands, the White Nile floods, and vast areas of the country become unreachable. Dirt tracks turn to mud. Most overland travel outside Juba becomes impossible. Even flights to secondary towns become unreliable.
The practical caveat is that none of this matters much if the security situation is bad. In recent years, armed conflict has been the limiting factor far more than weather. Check current advisories before making any plans, regardless of the season.