What is the difference between the two Structures:
1: VERB ni VERB. ex:sentaku ni dekakemasu.
2:VERB(te form) VERB. ex:nagarete kaeimashita.
(The example are from momotarou)
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Another phrase for "reason" is "in order to", which is usually constructed with (の)ために. The shorter version of that is (の)に.
watashi ha nihongo no shinbun wo yomu no ni jisho wo tsukau.
In order to read Japanese newspaper, I use a dictionary.
That's the grammar point used in your first sentence "sentaku ni dekakemasu". So, the longer form would be "sentaku suru tame ni dekakemasu". に usually has a feeling of a target, not only in the locational sense. If you would have said "sentaku shite dekakemasu", the meaning would become "I do the laundry, then I leave". As you can see, the te-form connects two sentences rather loosely.
The same applies for "nagarete kaerimashita" (note kaeri, not kaei!), which literally can be translated as "it flows and it comes home", which is obviously the same as "it flowed home". But that's completely different to "In order to flow, it came home"