A couple of general guidelines ...
First, be sure the systems you're looking at use Intel CPU's => the Intel Core-architecture CPU's perform FAR better than AMD CPUs. I presume that's the case already, but you didn't give specific model numbers, so I can't verify it.
Second, for the usage you've described there's no reason to buy a quad core system -- any good Core 2 based CPU will provide far more CPU "horsepower" than you need. As you've already noted, the dual core CPUs run cooler than the quad cores -- a Core 2 Duo consumes up to 65 watts; a quad consumes up to 95 watts (or more -- the high-end units are 120 watt).
I've built many quad-core systems for friends (typically using Q9450 or Q9550 CPUs); but selected an E8400 for my own use -- I like my systems to run cool and quiet ... with less heat being generated I can keep the fans at lower speeds & thus keep the system quieter (it's hard to even tell it's on). I could easily swap my E8400 out for a Q9550 (or better) ... but have NO desire to do so. I'm confident you'd have PLENTY of "horsepower" with a good Core 2 duo.
Having said that, a quad core does have a bit more "future proofing" potential -- clearly the more "horsepower" you have, the better you can deal with future, even-more-bloated, software. But to really evaluate the relative systems, you need to provide the model #'s of the systems you're comparing. To put things in perspective, a typical system of 4 years ago was probably based on a Pentium-IV at 3.2GHz. This scores 496 on PassMark's CPUMark -- an excellent measure of CPU "horsepower". A couple of typical Core 2 Duo's are an E7300, which scores 1728, or an E8400, which scores 2087. A typical Core 2 Quad, the Q8200, scores 3131. Note that most systems use a VERY small percentage of their CPU's "horsepower" during normal usage -- so the difference in day-to-day performance you'll see is much less than these numbers might imply unless you're doing very CPU-intensive tasks (which you indicated you are not). You might want to run Task Manager on your current system and observe the CPU % for a while, just to get a feel for how much you're using now. A faster CPU WILL, however, notably improve booting and program initialization times (where the CPU is heavily utilized).
Bottom line: A Core 2 Duo is PLENTY for your needs, and is what I'd recommend for you. A quad will provide very marginal gains and will both use more power and run warmer.