Are you saying the identical URL shows different sites in different browsers on the same computer? That's definitely not normal.

Is this limited to your computer, or if you have someone check the site in Firefox on another computer, does the same problem occur?

If it's just your computer: could you check for any customized connection settings? (Details below)

If it affects all computers: Does the site have any scripts, style rules, or server-level rules (e.g., in a .htaccess file) that might modify content for different browsers?

Could you check your Firefox connection settings here:

"3-bar" menu button (or Firefox menu) > Preferences

In the left column, click Advanced, then on the right side select the Network mini-tab at the top. Click the "Settings" button.

I'm not sure of the default setting on Mac, but you could try "No proxy" to see whether that works better.

In your Question Details > More System Details, you have a plugin with this description:

Aspera Web enables fast and secure file transfers using fasp. For more information, visit www.asperasoft.com.

Is there any possibility that plugin implements its own caching mechanism that could be affecting what Firefox retrieves?

I did what you suggested and am still having the same issue. The really odd thing is that when I went back to Safari it went back to the old site - which it didn't about an hour ago!

I've shut down, re-booted, cleaned bookmarks, history, etc. and still it takes me to the old site.

I created this site using Weebly.com. Everyone else seems to be able to see the site except me! Very frustrating!!

Is this a home network? You might try "power cycling" (turning off and on again) your router/cable modem to see whether that makes any difference since the memory of the old site might be "upstream" from your computer.

I just went to another computer in the house and used Firefox and it went to the new site. Not sure what to do at this point except know that the rest of the world can see my new site but I can't - at least not from my computer!

You can remove all data stored in Firefox from a specific domain via "Forget About This Site" in the right-click context menu of an history entry ("History > Show All History" or "View > Sidebar > History").

Using "Forget About This Site" will remove all data stored in Firefox from that domain like bookmarks and history and cookies and passwords and cache and exceptions, so be cautious. If you have a password or other data from that domain that you do not want to lose then make sure to backup this data or make a note.

Can't recover from this 'forget' unless you have a backup of involved files.

If you revisit a 'forgotten' website then data from that website will be saved once again.

Are you using a bookmark or are you starting with the main (home) page of this website?

If you use a bookmark or use a link that has specific GET data appended to the link or a page that has been requested by sending a form with POST data to open a specific page on the server then try to navigate to that page starting with the main page or the sign in page in case this bookmark/link is currently broken.

It's all working now but I didn't do anything other then what I had done before! Thank you all.