As Maxpm hinted at in his comment, it's mostly an issue of speed. A cable lift is faster than a chain lift, so the coaster gets to the top faster. I don't think there's any major gameplay reason to have one over the other, though a cable lift would likely shorten the ride time and thus might effect ratings. One would have to build two coasters that are otherwise the same to test the effects.
In terms of gameplay: It's been a while since I've messed around with Rollercoaster Tycoon 3, but if I remember right, cable lifts require a special start and end piece, and requires a completely straight lift hill. This means you need a certain amount of horizontal space to be able to use it successfully, and ideally, it should be used for the main starting hill. I'm not 100%, but you might be limited to only one cable lift start/end point. Chain lifts also only work on straight inclines, but you can add tight turns between them to break up the lift hill and make it spiral (though you need a certain train length to be able to do this). So chain lifts work well in areas with space limitations.
Outside of the game: Cable lifts are used in a variety of coasters and (according to wikipedia) they require less maintenance, and are fairly quiet compared to the loud, clanking chain lift. The earliest coaster to use a cable lift (at least, per the wikipedia article) was built in 2001, though the method was likely used prior to that. Rollercoaster Tycoon 3 came out in 2004 (with RCT2 coming out in 2002, and RCT1 coming out in 1999). Likely, the cable lift in RCT3 was included for realism and ride variety as opposed to any major gameplay reason.