Are you looking at laptops or what?
Dual-core means the number of processor cores working in there (two processors on one chip).
Your old processor is probably a single 2.3GHz chip. A 1.86GHz dual means you've got two cores each working at 1.86GHz. Software that can use two processors will run much much faster on a dual-core system. And even single-threaded software will usually gain some advantage from dual core processors because one processor works exclusively for the software running while the other handles all of the Windows stuff in the background.
Not to mention that architectural changes have occured so today's chips do a lot more work clock-for-clock compared to older generations of chips. For example, at the launch of the Core 2 Duo chips, the new e6300 (2x1.86GHz, 2MB cache) could match or even beat the fastest Pentium 4 dual-core ever made (P4D 965 EE - 2x3.73GHz, 2x2MB cache) even though it had exactly 1/2 the speed & cache.