It’s a matter of emphasis. The OP’s two examples would be typically found in speech, not writing, and the way in which they were spoken would reinforce the meaning. For example, we might expect The dog ISN'T running to be in response to someone who said Hey, stop your dog running like that. On the other hand, The dog's not RUNning could be followed by There must be something wrong with him today. I recognize that these are not the best examples to illustrate the point, but I hope they will give some idea of the difference. The first emphasises that there’s something the dog isn’t doing, while the second emphasises what it is that the dog isn’t doing.
As for the propriety of using contractions in writing, Pam Peters, in ‘The Cambridge Guide to English Usage’ takes a typically pragmatic view:
The writers of formal documents may feel that [contractions] undermine
the authority and dignity of their words. But the interactive quality
that contractions lend to a style is these days often sought, in
business and elsewhere. They facilitate reading by reducing the space
taken up by predictable elements of the verb phrase, and help to
establish the underlying rhythms of prose.
It’s perhaps worth adding that French uses contractions even in the most formal contexts. There are no non-contracted alternatives to je n’ai pas and so on. And of course there’s one quasi-contaraction that is always found in English, the singular possessive apostrophe -’s.