When the solder melts for more than a second of two, a layer of tin oxide forms at its surface. The longer you heat the thicker the crust and the toughest it is to get a good joint. A crusty surface makes it hard to inspect too. People will frequently add more solder to make it look good again. What makes it look good again is the flux in the solder, but doing this also adds solder, and the joints will soon look fat and may short the adjacent pins. But you can just add flux alone, and you will get the same result without adding solder material.
I'm no chemist but the flux sort of dissolves the oxide layer that forms at the surface of the solder material, be it on the pins or on the pcb pads. Thus it reduces the risk of making cracked solder joints.
Restoring the wettability (surface tension) of the solder is also very useful when you have to repair shorted leads. When you add flux over the shorted pins, you just have to touch the pins with a blade-shaped soldering tip and the solder gets sucked by the surface tension of the (clean) tip. No way that you can do that without flux. I do many, many TSSOP and TQFP packages. Desoldering wick to repair shorts? Never! I don't need it! The flux technique does much much better, and you do not overheat the pads nor the IC pins.
itsmoked, does Weller make blade-shaped tips for this iron? If they do, give it a try and your only regret will be that you did not try it earlier.
One comment about the water-soluble flux, you must ABSOLUTELY clean the board thoroughly with hot water (60C is hot on our hands but not for an electronic part) after you're done. This flux is very corrosive and it attacks metal. I've seen boards scrapped because of water-soluble flux remains under BGAs.
I use a 5-45X binocular to inspect my works, and I can tell that with flux usage the quality of the solder joints is as nice as what comes from an assembly house. My prototypes can be as reliable as the production boards. I can also tell that my touch-ups look better than those made by the people at the assembly house.
The fine tweezers are in my toolset too, as well as a dentist pick. It does less damage than an Xacto, which I use to cut traces.