There shouldn't be any difference between the two so if one doesn't work, the other one shouldn't either. I took a second look at those pages and If my assumptions are correct about the .asp e-mailer, they may be telling you that they don't support the <cfmail> tag since it must use a mail server for delivery. The .asp objects may have built in support for mail transport and may not need the external mail server. This means your mail scripts may have to be written using the MS CDonts package. Here's a piece of untested code that might work for you if they have CDonts installed. CDonts is the standard MS e-mail component for .asp and is probably available on your server.
<cfobject
type="com"
action="create"
name="myEmail"
class="CDONTS.Newmail"
>
<cfset myEmail = myEmail.Subject="My Subject">
<cfset myEmail = myEmail.To="recipient@email.address">
<cfset myEmail = myEmail.From="sender@address.com">
<cfset myEmail = myEmail.Body="This is my message">
<cfset myEmail = myEmail.send>
If you don't want to use this, you may have to find a CF host that supports the <cfmail> tag. Any host that basically told you to go look at external links to get your e-mail working isn't providing you with adequate support though IMO. I would expect a little more help in getting such an important part of CF working.
<shameLessPlug>We of course provide this with our CF hosting

</shameLessPlug>
Hope this helps and let me know if that code works for you,
GJ