There are two reasons:
- to secure the data against the (possibly unlikely event) that the company which performed the original analysis ceases to operate or loses your data.
- (more important) to import it elsewhere, and so increase the potential number of matches that you make and/or access analysis tools that may not be available with your initial 'provider'.
As https://genealogy.stackexchange.com/a/3550/104 says, anyone considering uploading their data to a public DNA database or a DNA utility site such as gedmatch should ensure they are happy with the privacy arrangements first.
Family Tree DNA has a Transfer Program that allows you to import DNA results from both 23andMe and AncestryDNA. It might be useful to have another company analyze your DNA to see if there is any significant difference in what they say are your ethnic origins. But the greatest value would probably be in having your DNA match against the other Family Tree DNA customers to find potential relatives. DNAeXplained shares some other possible benefits based on additional features that Family Tree DNA offers for genealogists.
Family Tree DNA allows you to join specific groups for DNA comparisons. In joining a family surname group, you may find other submissions that closely match your submission. You can contact close matches via the FT database, and in corresponding with a match you may find out how you are related or find and unknown relative who has your family line identified back further than you do. When you contact them, you are not given their email address, if they want to respond to you it is up to them. Same for you. Joining a haplo-group that matches yours may tell you where your family originated even if you can't find the links between.
Here's one use of exported raw data from Ancestry.com.
I just used my data from Ancestry.com to compare the results to various interesting rsIDs related to health, appearance, etc. using Promethease:
There's a $5 online version, $2 offline (Mac or Windows, takes about 10 minutes to run), or free offline (same program slowed down to 4 hours).
The offline version means that while it downloads an updated database of interesting SNPs, it does not upload yours back to anywhere, all of the comparison happens on your own computer.
While it's not a diagnosis tool, seeing how my genes stack up to current research into risk factors for various diseases is very interesting and well worth the few button clicks it took to get it from Ancestry.