Can't send emails

I'm presently using a laptop with Windows 7 and Windows Live Mail but I have bought a new laptop running Windows 10. On this laptop I have installed Thunderbird to handle my emails. I have 4 separate email accounts and I have set them up on Thunderbird and it seemed to be working fine as my emails were being downloaded when I started Thunderbird but I was still answering my emails using the Windows 7 laptop. I decided to run a test tonight on the Windows 10 laptop by sending test emails from one account to the others but although they appeared to send and no error messages appeared no messages appeared. I have set up both laptops side by side and checked the setting etc. and everything is correct. The only difference I can see is that although my incoming mail is on both laptops, my outgoing mail has my name in front of in Thunderbird and there doesn't appear to be a way to remove my name. By the way as well as checking and double checking all my settings I have also switched off my antivirus and my firewall and sent test emails and nothing happened. I'm beginning to wonder if Thunderbird is worth the bother as getting to the stage I'm at now has been a nightmare. Any time I switched email programs in the past a couple of clicks and it was done. With Thunderbird it took be most of Sunday afternoon just to get my email contacts imported! My only reason for picking Thunderbird was because everyone said it was great and it looked similar to Live Mail. So can anyone help me out here as to why I can receive but not send emails?