Would you consider it a notable difference if one distribution has a negative mean, and the other has a positive mean? This difference may be critical in some situations.
Let $X$ be the constant $-1$. $E[X] = -1.$
Let $Y$ be $-1$ with probability $99.5\%$, and $399$ with probability $0.5\%$. You can think of $Y$ as the net value of a raffle ticket you bought for $\$1$ for a $\$400$ prize. $E[Y] = 1$. If you take gambles like this repeatedly, you will win in the long run with high probability.
These are not continuous, but you can make slight adjustments to make them continuous. Let $X'$ have density $1/2$ from $-2$ to $0$, and let $Y'$ have density $1/2$ from $-2$ to $-1/100$ and from $400-1/100$ to $400.$ These exhibit the same behavior.
More subtly, you can make the difference between a winning and a losing gamble occur between, say, the $55$th and $60$th percentiles. Suppose you risk $3$ to gain $4,$ e.g., in a poker game you call a $\$3$ overbet into a pot of $\$1$ on the last round. If you lose at most $4/7 \approx 57\%$ then you have a winning gamble. The $5\%$ quantiles can't see the difference between losing $56\%$ of the time (a good gamble) and losing $59\%$ (a bad gamble). This example can also be made continuous.