I do forms for several different websites. Would it be better to open a different account for each client

It is up to you. If you want to manage all your forms in one account. If you want to be organize you can create new account for each of your client. 

You can also organize your form in one account, here's a guide: How-to-Add-or-Group-Your-Forms-into-Folders

Keep in mind that you can request form transfer to any account. 

Hope that helps. If you have any other questions, please let us know. Thank you.