Eutectic solder refers to a mixture that goes directly from liquid to solid. Non eutectic solders have a range of temperatures in which some part of the alloy solidifies or melts and other parts don't resulting in a plastic phase where the material is soft and pliable--i.e. plastic.
IMHO, the difference (between 60:40 and 63:37 tin/lead) is that if a part is moving because you're holding it by hand, a eutectic solder could lead to occasionally less secure joints that look good, while a solder with a plastic-phase could lead to occasionally more reliable joints that look bad. Neither of those cases is a good thing, both being bad joints relatively speaking. Personally, I'd rather have the bad joint and check them than have bad looking joints that I redo as a matter of course, so I use eutectic, but this is because most bad joints with eutectic will look absolutely horrible, and many will feel weak. I feel it gives me more reliable results. For a beginner, it might be better to have the visibly bad looking joints, even if a lot of them are perfectly alright.
Maybe someone could say that better. Thing is, eutectic hardens instantly, so any movement leaves the part free to move. The plastic phase partly hardens, so the plastic phase stuff can look bad even if the part-being-soldered was stable (not moving) when the solder actually hardened, which would give a reliable, but ugly joint. You might end up remelting this joint as a matter of course.
If the parts are perfectly stable through the full plastic/hardening phase, you'll get good joints with either. In that case, you can still get occasional ugliness with eutectic, with wavy or spotted surface, instead of shiny. It's a consequence of the quick drying. Another reason I don't recommend eutectic for others, but still prefer it for myself.
Maybe one more try... Any movement at all will show up as waves or ugliness with 60:40, even if the joint is perfectly alright. With eutectic, not all good joints will be shiny, but most movement (during that short 1/2 second or so) will cause a bad joint.
Temperature isn't significant for for the two alloys mentioned here. The melting temperature for 63:37 is the same as the plastic temperature for 60:40, and the melting temperature for 60:40 is only 5C higher. The temperature difference for other formulations is more significant, so temperature considerations enter into it. Some of these have a large temperature difference between melting and plastic phases. I'd just stay away from any silver-bearing or unleaded solder that's not eutectic. Yet another reason I prefer to just learn to deal with eutectic and be done with it. Still, I have to admit 60:40 is more practical than 63:37 if that's all you ever use.
Edited by SiBurning - 7/7/12 at 10:20pm