Paper weight and thickness may have little or no noticable effect on printed image quality. However, heavier, thicker papers feel more substantial and can lend visual weight to a project.
Heavy weight paper can lend an aura of importance and seriousness not found in flimsier products. This is especially true in business cards. Often thin paper screams cheap desktop printing, unless you're designing a mail order catalog or directory which tend to work better with thinner, lightweight papers.
Thicker papers (especially coated papers) hold up well when there is heavy ink coverage. For full color photos from your desktop printer thicker inkjet photo papers are best. However, some desktop printers may not handle thicker papers well, especially if the printer doesn't have a straight paper path. Thicker papers also work best for diecuts.
For desktop and offset printing choose a paper that feels right to you and will work with your desktop printer or the commercial printing process you choose.
More ways to Design With Printing in Mind that can save time and money.