"The man is drinking water."
Translation:Mannen drikker vann.
The difference is what is in common use in bokmål and the so-called standard øst-norsk dialect.
Although both vann and vatn are accepted as written forms in bokmål, vann is the most and commonly used one. vatn has more in common with the dialects so you will hear people saying it that way too, but probably less written that way. In nynorsk vatn is the only acceptable spelling.
If you want to look up the official way a word can be spelled and inflected in Norwegian the (online) dictionary given out by Norsk språkråd is the place to look:
bokmål and nynorsk are just written forms. Hardly anyone talk like that all the time.
Standard øst-norsk that the computer voice uses (more or less successfully) is most likely the closest you will get to a spoken form of bokmål. (Unless you bump into fanatics ;-)
Based on what pupils choose, roughly 4/5 of the population prefer to use bokmål when writing. The rest prefer to use nynorsk. (Goes a bit up and down each year: http://www.ssb.no/utdanning/statistikker/utgrs/aar/2014-12-12?fane=tabell#content )
Bokmål (transl. Book tongue) and Nynorsk (transl. New Norwegian) are two written standards of the same language - Norwegian (norsk). People usually start with learning Bokmål because it's used by about 90% of the population while Nynorsk has a solid base in the western part of the country.
When it comes to pronunciation, that's a totally different story. Although there is a trend to standardise it, it's not quite there yet. This means that everyone is allowed their own twist when pronouncing words and that is influenced by the dialect they speak. As @noko_heilt_anna had already said, this woman bot mostly tries to speak in Standard East Norwegian which is the unofficial spoken standard of Bokmål. Funny thing is that even native speakers will have some minor difficulties when meeting a fellow Norwegian who is not from their county.
All in all, although these standards and variations make it harder for learners to fully grasp the language, I think it's also great proof of how rich and diverse languages can be.