They look similar but are completely different products.
Curd is made at home by adding a spoonful of old curd or lemon juice to warm milk and letting it ferment naturally at room temperature. This uses whatever natural bacteria is floating around so every batch varies.
Yogurt is made commercially using specific bacteria strains like Lactobacillus and Streptococcus under controlled conditions. This ensures consistent taste, texture and probiotic content.
Curd has a softer grainy texture and mild taste while yogurt is smooth and creamy and tangier. Yogurt is better for people with lactose intolerance because controlled fermentation removes more lactose. Yogurt has higher and more specific probiotics for gut health.
Curd is more traditional and cheap to make at home. Both are nutritious but yogurt offers better probiotic benefits while curd is your grandma's classic choice.