Try this (usually at this point I'd have two versions, HTML and plaintext):
<cfsavecontent variable="content"><cfinclude template="content.cfm"></cfsavecontent>
<cfmail to="#user.email#" from="email@example.com" subject="Message from #sender_name#" type="html">
Also wrap content.cfm inside <cfoutput></cfoutput>
Your code works in cfmail because Coldfusion performs cfoutput implicitly within the cfmail tag. The solution is therefore to add the cfoutput tag in content.cfm, as follows:
Incidentally, your post has a typo: cfinclue instead of cfinclude. This is just a typo to get out of the way, and has nothing to do with the issue.