It depends on the country but in the US if you don't include the suffix LLC the counter-party in any dispute CAN make the argument that your liability 'shield' should not hold (granting access to your personal assets) because you implied you were simply a sole-proprietorship.
To be safe, use the LLC in emails. ALWAYS use LLC in contracts.
One good rule is to follow german laws here - they make sense.
On every communication have a footer (on emails, or on nomal letters as part of the letterhead) giving your full name and registered address, registration location and code (i.e. where and under what number), plus minimal legal contact details (an email, a phone number). And the name of the general manager / director / whatever it is in your jurisdiction. Plus your website ;)
This way you look professional, people always know whom to deal with and there is no question about your legal form etc.
I am always astonished by the number of (primarily american) smaller companies where I have to start searching for registration information. Does not look professional. You "pretend" to be a LLC, please be so kind to provide me with the information I need to validate that claim.