Both mean "to trust".
But which is more used in speaking?
Both mean "to trust".
There is a slight danger here in assuming that both words are perfect synonyms. They are not. You can always exchange trauen for vertrauen insofar as both mean "to trust". But you cannot exchange vertrauen for trauen insofar as trauen can have some other meanings like to dare, to wed etc. Also das Vertrauen might be a capitalised noun, but trauen like in Menschen trauen sich is always a verb.
Looking at the books corpus the evil search engine has accumulated the case seems quite clear:
But this is disregarding the aforementioned possible dissimilarities and just counting words in books. Assuming you are only interested in usage frequency for equivalents of "to trust" the situation is much more difficult.
Looking at a German newspaper corpus based on material crawled in 2011:
Word: vertrauen Number of occurrences: 4,260 Rank: 8,285 Frequency class: 11 See also: Vertrauen, VERTRAUEN Part of speech: Verb
Baseform of: vertraute, vertrauten, vertraut, vertraue
Part of: vertrauen auf, Kunden vertrauen, vertrauen schenken, vertrauen mit, auf etwas vertrauen, auf jemanden vertrauen
Synonym: trauen, erweisen, anvertrauen
Word: trauen Number of occurrences: 4,241 Rank: 8,311 Frequency class: 11
See also: Trauen
Part of speech: Verb
Baseform of: trauten, Trau, traut, traute
Part of: nicht trauen, sich nicht trauen, nicht über den Weg trauen, sich etwas trauen, sich trauen lassen, seinen Ohren nicht trauen, seinen Augen kaum trauen, Seinen Augen nicht trauen, seinen Ohren kaum trauen, seinen eigenen Augen trauen
Hyphenation: trau|en Description: Glauben schenken Synonym: vertrauen, heiraten, getrauen, riskieren, wagen, unterfangen, verheiraten, verehelichen, unterstehen, ehelichen, erdreisten, vermählen, verlassen
I conclude from these numbers that vertrauen has an advantage, but that the difference in actual usage is not so big.
If this question is meant to inquire about which verb would sound more natural when speaking, then it seems much more important check for two other things:
- Possible double meanings. These are unlikely to occur with vertrauen.
- More frequent pairings of words (collocations). If "you do not trust your own eyes, it would sound sightly weird to use "Du vertraust deinen eigenen Augen nicht,". That would mark you as a non-native speaker or someone with a special philosophy (esotericism, buddhism, jedi…). More natural in most of these cases involving eyesight would be to choose "Du traust deinen eigenen Augen nicht" .