Hardware definitely has an effect on animation performance but the Flash player can deal with a surprising amount of information before it begins to slow down noticeably so you can usually up your framerate to get a smoother look to your movies (I find I can usually get away with at least 30 fps without too much bother).
If things are playing slow here are some things which may help...
1) If you're using a lot of bitmap images (like imported jpegs etc) try and get them down to the smallest file size you can. In some cases you can use >trace bitmap to convert them to vector images which are more efficient and therefore faster. This sometimes backfires though and really complex images are best left in their original format. Try pushing up the contrast in Photoshop too which drops a bit of detail and helps to reduce filesize, a very slight blur (1 pixel) can be added too which makes a big difference to jpeg filesize without compromising your image greatly.
2) If you're using vector shapes they draw faster if they don't have a stroke outline.
3) If you're playing with the alpha transparency settings things will visibly slow down when you animate multiple images. This is because Flash has to work out what's on the background layer, then what's in the foreground and perform calculations to mix the images: much harder than placing one image directly over another. If you're animating against a black or white background use >brightness to simulate fades because it's much more efficient.
4) Gradient fills take longer to draw than solid colours.
5) Macs are weird about framerate - you have to set it at least 31fps to get smooth performance.