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Speaking "double Dutch" simply means to talk in gibberish. Any time a person has no idea what a speaker is saying, the person might say "It sounds double Dutch to me." or "He's speaking double Dutch."
It is not used much any more. It comes from the fact that, even though Dutch and English have the same common root language, most English speakers find Dutch incredibly hard to understand.
"Speaking in tongues" is also used when someone is speaking gibberish, but it is usually reserved for a religious context. In some churches that promote extreme emotional outlet in their services (Pentecostal is an example), some people will find themselves "possessed" of some holy spirit and start speaking in "languages" which don't actually exist.
"Double Dutch" could be used in any situation where you don't understand anything a person is saying. "Speaking in tongues" would almost always have some religious connection.
