1. Active matrix typically will have deeper, more vibrant colors (contrast ratio)
2. Active matrix has a wider field of view (the angle of the observer to the screen is not as critical as for a passive matrix screen).
3. The active matrix screen is faster responding.
4. The passive matrix screens cannot display as many colors.
5. Passive, dual scan screens have a division in the screen.
   

So what does this mean? If you can't tell the difference between them as far as color goes, then you can disregard item 1 above.

If you will have people looking over your shoulder, item 2 is critical since the angle that they have to the screen will almost certainly be non-optimal for a passive matrix screen. So if you frequently want someone else to look at your screen (say if you're doing a sales demo without a projection device), you absolutely need the active matrix. On a positive note, the passive matrix pretty much makes people sitting next to you in meetings unable to decipher what you are typing without being obvious about it if you have a passive matrix.

Response speed becomes important when something is moving across the screen quickly. The most common instance of this is when you move a mouse cursor. On a Passive screen, because of the slow response time, the pixels that compose an image of the cursor never get a chance to get black before they're told to go back to white. So until you slow down or stop, the cursor goes invisible. With an Active screen, you can see each fleeting image of the moving cursor, which helps you track it, improving your productivity since you don't spend so much time looking for the cursor.

The largest number of simultaneous colors that most passive matrix displays can handle is 256. On an active matrix display, up to 16.7 million colors can be displayed simultaneously (depending on the current screen resolution). If you're really into graphics and you want near photo-realistic display , you'll need an active matrix display. Some late model passive matrix screens are able to display more than 256 colors, but still not as many as the active matrix version.

Finally, since the passive screen is slower, the screen is physically divided into two pieces which are scanned simultaneously. This results in a faint line across the center of the display where the two screen halves join. Some people find the line rather distracting, others do not notice it.

Active matrix is a lot more expensive, and because of manufacturing issues, is also typically in short supply (so you may have to wait for a while to get a machine with it).

There are some other issues. Active matrix displays may occasionally lose pixels (permanently). So you may have a tiny green (or other colored dot) show up someplace on your display. Most manufacturers will replace the screen if more than a certain number of pixels fail. This happens more rarely with passive matrix