The answer surprisingly is no.
Metal hardness is measured by what is called the Vickers scale where harder metals receive a higher Vickers score than less hard metals. 9kt has a Vickers hardness of 120 and 18kt has a Vickers hardness of 125. This shows 18kt as harder, but the difference is so slight that in practical terms 9kt and 18kt are much the same.
Yellow gold, white gold and rose gold are all measured using the Vickers scale, and as the same carat of yellow gold has the same gold proportional content as white gold and yellow gold, each of these metals has the same Vickers score value (that means that 9kt yellow gold, 9kt rose gold and 9kt white gold each have a Vickers score of 120 whilst 18kt yellow gold, 18kt rose gold and 18kt white gold has a Vickers score of 125).
There is another difference that should be considered when comparing 9kt and 18kt. 9kt is more difficult to bend and is a little more springy and therefore a fine 9kt ring may be less likely to bend out of shape than a fine 18kt ring. However, if the ring is of a good sturdy construction neither metal will be likely to bend out of shape.
18kt yellow gold is more yellow than 9kt yellow gold with 18kt yellow gold having a richer golden color making 18kt yellow gold a more popular choice of yellow gold than 9kt for ladies engagement rings.
9kt rose gold has a darker rose-copper color than 18kt rose gold which has a more subtle rose color.
9kt white gold and 18kt white gold have a similar white color (as normally both have a rhodium plating finish (see white gold vs. platinum information page).
In practical terms, 18kt generally holds up to everyday wear better than 9kt and tends to look better than 9kt as it ages over the years. This is one of the reasons why 18kt is normally preferred for ladies engagement rings rather than 9kt.
18kt is the superior metal for lady's and men's rings provided that the ring is not a lightweight construction (Our rings are a good construction). However, 18kt is almost twice the price of 9kt, which is usually fine for ladies engagement rings, though in a large men's ring the 18kt price can be a noticeable difference.
At Gillett's we believe that for a wedding ring that you wear every day of your life is worth the difference and we accordingly recommend that 18kt be selected whenever possible.