We construct biharmonic non-harmonic maps between Riemannian manifolds $(M,g)$ and $(N,h)$ by first making the ansatz that $\varphi :(M,g)\to (N,h)$ be a harmonic map and then deforming the metric on $N$ by $${\tilde{h}}_{\alpha }=\alpha h+(1-\alpha )df\otimes df$$ to render $\varphi$ biharmonic, where $f$ is a smooth function with gradient of constant norm on $(N,h)$ and $\alpha \in (0,1)$. We construct new examples of biharmonic non-harmonic maps, and we characterize the biharmonicity of some curves on Riemannian manifolds.