What is the difference between TrueType fonts and Type-1 fonts?

The Postscript Type-1 specification was created by Adobe back in 1985 or so. Type-1 fonts are vector based. You can find the specification in "Adobe Type 1. Font Format.".

TrueType fonts were defined by Apple a couple of years earlier so True Type and PostScript were competitors in the 1990s. Microsoft picked up True Type for the native Windows font format in the beginning 1990s (for using PostScript, additional tools like Adobe Type manager were necessary).

Today, Microsoft is fading out support for PostScript fonts. Try using one as an UI font in Vista. Good luck ;-)

As a successor of TrueType, Microsoft (I think together with Adobe) created the Open Type (anytime around 2000) format and Adobe converted their whole font library into the new format (you can still get them as Type-1 fonts).

A very key difference is that PostScript (and PostScript flavoured OpenType) supports cubic Bézier curves, where each arc of each glyph is described by four control points. TrueType (and tt flavored OpenType) uses quadratic curves, with each arc having only three control points. This offers less control over the shape of the curve.

Another key difference is the way they perform hinting. Since TrueType was originally targeted to low resolution screen rendering, its hinting system works by adjusting the curves to fit nicely on pixel lattice points, using a fairly elaborate bytecode mechanism. PostScript fonts were intended for higher resolution paper prints, and used guidelines to snap curves to right angles at appropriate places.