We consider the conjugacy classes of diffeomorphisms of the interval, endowed with the $C^1$-topology. Given two diffeomorphisms $f,g$ of $[0; 1]$ without hyperbolic fixed points, we give a complete answer to the following two questions: under what conditions does there exist a sequence of smooth conjugates $h_n f h_n^{-1}$ of $f$ tending to $g$ in the $C^1$-topology? under what conditions does there exist a continuous path of $C^1$-diffeomorphisms $h_t$ such that $h_t f h_t^{-1}$ tends to $g$ in the $C^1$-topology? We also present some consequences of these results to the study of $C^1$-centralizers for $C^1$-contractions of $[0;\infty)$; for instance, we exhibit a $C^1$-contraction whose centralizer is uncountable and abelian, but is not a flow.