When we talk about "redeeming" a first-born son or a captive, we use "pidyon" (root פדה). When we talk about what God does for us we use "goel" (גאל). My dictionary translates both of these roots as "redeem" with no further differences. Are they synonyms (with customary usages) or is there some deeper difference between them?
Malbim (to Ps. 69:19) draws the following distinction:
גאל means to redeem someone or something because of your relationship. (The cases in Lev. 25 where a person has to sell his property, or even himself into slavery, and is "redeemed" by himself or a close relative, are all described with this root.)
פדה means to redeem someone or something because of its intrinsic value.
These definitions are taken from Etymological Dictionary of Biblical Hebrew, compiled by Rabbi Matityahu Clark, based on the teachings of Rabbi Shamshon Raphael Hirsch. (Feldheim, 1999)
גאל means "release" or "purge;" more specifically, to (1) "redeem from existing danger," [Shemot 6:6, Bereishit 48:16] (2) to "ransom," [Bamidbar 35:12] (3) "pollution requiring cleansing," [Yeshayahu 59:3] or (4) "redeemer" [Vayikra 25:25, Bamidbar 5:8].
Cognate meanings of the word include "activate/deactivate," as seen several related words.
פדה, on the other hand, means to "liberate; extract from control." More specifically, (1) "redeeming" [Shemot 21:8, Bereishit 9:27] or (2) "deliverance" [Shemot 8:19].
One cognate meaning of the word is given: to "react to externals."
The difference then, I suppose, would be that גאל is used to mean "saving," where there is some existing negative force (danger, pollution, etc); whereas פדה is used in cases where a party is saved from another's control.