Ok from a standpoint of someone who uses a dual core (Galaxy Tab 7 Plus - same chipset as Galaxy SII) and a single core (Galaxy SL) at the same time:
In the more "hidden" type of difference, the Tab appears to have better battery life. Consider that the tab has a screen more than twice that of the phone (both are using LCD), a more powerful processor, and a little over than twice the battery size (phone has 1650mah, tab has 4000mah), one would expect that they would have similar battery life (2x screen size, 2x cores, more powerful GPU, battery difference). But they don't. In theory a dual core processor can save more battery than a single core processor since it finishes tasks faster. Both are setup to have the same push notifications and whatnot. Basically the Tab is like a clone of my phone (both are 3G devices with unlimited data). Leaving both at night and with no calls, the phone still drains way faster in terms of percentage.
As for the OP's actual needs:
Depends on the website whether it will load faster or not. A mobile site will load at the same time, since a dual core will only use the second core if needed. If its not needed it will run single core. A website with lots of Java or Flash will load faster probably on the dual core, but I'd think the speed difference will rely more on the network speed than the phone itself.
As for multitasking, you will be able to multitask more smoothly on a dual core. You are limited by two things in multi-tasking: cores and RAM. More RAM allows you more apps in cache, more cores and processing speed allows you to throw more resources at an app to keep it running smoothly. For example, on two theoretical devices with the same amount of RAM and clock speed (with the difference one is dual core). Lets say both devices can hold 4 apps simultaneously on its RAM. The single core device would have to divide its CPU cycles in a single core to the 5 apps, basically each app will be running with 24%CPU power behind it. On a dual core device, each app can have as much as 50% CPU power behind it, or half of each core for the four apps.
Basically, RAM will dictate how many apps you can have running in the background, but the cores and clockspeed will determine how smooth they'd be running.
DISCLAIMER: All examples given are overly simplified. There are more factors than just cores. Take for instance how the dual core Snapdragon S4 has the same performance as the quad core Exynos 4412 despite having only half the cores.