I could (but won't) claim, as one who always* changes a new car's tires to wider, lower profile ones (my preference), that:
1) the ride will become a little bit stiffer (you're only going from 70 to 60 aspect ratios, which isn't much).
2) the steering will lose a little on-center feel (because of increased width), but will also become much more responsive and more precisely follow your steering input (again because of stiffer sidewalls, a consequence of shorter sidewalls as the aspect ratio decreases).
As for noise, that depends almost entirely on the design of the tread, and it's regardless of tire size attributes.

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* On my 240's, they wear 205/55-16 (on Hydra) and 205/55-15 (on Virgo) Michelin Pilot Exalto PE2 tires. On my daughter's new ('07) S60 2.5T, I had her original tires (205/55-16 Michelin MXV4) replaced with 225/50-16 Exalto PE2 tires as well. I just can't tolerate "4-season" or "touring" tires (the classification of MXV4's) -- but, of course, it forces me to also put 4 snows on my cars in winter :-).
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But those generalizations assume that all other factors are equal between your 70 and 60 series tires -- they usually aren't.

Seventy-series tires are usually intended more for the "soccer mom" and grandparents cars (priority being soft, quiet ride, long-lasting, and no expections of good handling or braking); while 60-series tires are more performance oriented. The 70-series tires are often set apart and meant for an altogether different market.

Usually, the major tire manufacturers will, with only some exceptions, have tinkered in many ways other than mere size (sidewall plies, construction of the belt, design and compounds in the tread, etc.) so that there's little direct similarity between the 70- and 60-series tires.

You'll find more similarity between 60-, and 55- (sometimes even 50-) series tires than between 60- and 70-series.

So you may find that there's a bigger difference in the tires than you should expect solely based on slight size differences.

You really have to research the manufacturers' descriptions. But at least I've cautioned you.

Good luck.