Walkden, 24, landed two well-timed head shots as she beat Morocco's Wiam Dislam 7-1 to win her +67kg bout.
It is Team GB's third taekwondo medal in Rio after Jade Jones claimed gold and Lutalo Muhammad won silver.
Cho, 27, lost 5-4 in an aggressive fight against Brazilian Maicon Siqueira in the men's +80kg.
Cho beat Iran's Sajjad Mardani 4-3 in the quarter-finals but lost his next bout to a last-minute head kick from Azerbaijan's Radik Isaev.
"I gave it my absolute everything but it's just disappointing not to come away with something from all the hard work me and my team have done together," Cho said.
"One of us had to lose and unfortunately that was me."
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Walkden, meanwhile, beat London 2012 champion Milica Mandic of Serbia 5-0 but lost 4-1 to China's Zheng Shuyin in a golden-point round in the semi-finals.
"I came here for gold but my coach said to me you have to go out there and fight for it," Walkden said.
"It's still an Olympic medal. In Tokyo I'll be trying to go a little bit further and try to get gold. It's only four years away."
Zheng went on to win her gold-medal bout 5-1 against Mexican Maria Espinoza while Isaev beat Abdoulrazak Issoufou 6-2.
Issoufou's silver meant Niger became the 87th nation to win a medal in Rio, a new Games record.
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Great Britain's Bianca Walkden won Olympic taekwondo bronze in Rio, but compatriot Mahama Cho narrowly missed out on a medal as he finished fourth.