No, it definitely doesn't not mean you can clock it twice as fast.
And, most of the time, you will get a speed up less than 2x. (And, no you don't get that from doubling the clock rate, you keep the clock rate the same.) Still, you won't have enough work to use both cores, or worse all 4 hyperthreads if it is a core that supports hyperthreading. So, it is generally faster than a single core chip at the same clock rate, but how much depends upon your workload.
And, incredibly enough, under the right workloads and the right configuration, you can sometimes make the dual core more than 2 times faster than a single core version. However, most of this is covered by the phrase: lies, damned lies, statistics, and benchmarking. I know because I used to do benchmarks for Intel's (network processing group), so we created benchmarks for various chips under development doing various networking tasks, e.g. acting like routers or web servers, etc. Depending upon what we wanted to show, we would often pick different chips to do the demonstration. It wasn't a lie, but it was very carefully tuned. Not the kind of workload you as a home user are likely to get, but maybe appropriate for a use-case Google, AT&T, or Cisco might have in mind.
And, by the way, there are sadly cases where the other side of that equation is true, where a dual core chip is actually slower than a single core chip. Fortunately those cases are also rare.
But whichever case applies, if you are a home user, you can assume that generally, you are getting some speed up, probably not quite 2x, more like 1.2-1.8x, but some. And, more importantly, it probably gets better when the system is under load. The more you push your computer to do many things at once, the more likely you get better performance out of a dual core machine.
And the number of cores that exhibit a performance boost is also generally limited. You don't get another 2x boost by going to 4 cores (except in particularly limited cases), much less a third 2x by going to 8 or a fourth by going to 16. Yes, it is possible to contrive such cases, but they are definitely not the norm.
So, my 4 core Atom netbook is still noticeably slower than this Xeon laptop and it isn't due to core count or even clock frequency. The speed at which your computer runs is much more complicated than that. But, if I have only one website open, the netbook seems to run just as fast at this laptop because the real limiting factor is network bandwidth, not computer speed at all.