Andrej Bauer points out that predicative constructions are more explicit and give more useful computational information, than impredicative ones.  This has two further consequences that are of interest. I want to point these out as an answer to the implicit question asked by "alephomega".

1. If a theorem about reasonable objects can be proven predicatively, this gives important information on the consistency strength (i.e. the proof-theoretic ordinal) associated with the theorem. Conversely, we know that some theorems such as Kruskal's theorem cannot be proven predicatively because these theorems lead to proof-theoretic ordinals that are too large.  Thus the mathematical analysis of predicativity <i>reveals</i> the full complexity of results such as Kruskal's theorem, rather than obscuring it.  This is separate from the philosophical analysis of predicativity. 

2. The "computability" consequences of a predicative proof can be extremely important. In the context of a countable group $G$, a "top-down" construction such as "intersect all subgroups of $G$ that contain the set $X$" will only naively give that the constructed object is $\Pi^1_1$ over $G\oplus X$. A bottom-up construction will usually show that the constructed object is actually arithmetical in G. Examination of the bottom-up proof can then give explicit bounds in the arithmetical hierarchy on the complexity of the constructed object. For example, in the case of a subgroup of a group $G$ generated by a subset $X$, the degree of the subgroup is no more than the Turing jump of $G \oplus X$; this is an enormously better bound than the naive $\Pi^1_1$ result.