Use the HDMI for the HD monitor. Of course, you can always try them out either way to see if there's a difference.
 
Just MHO but it's always nice to have both monitors the same resolution, but they don't have to be, you just won't have the same screen coverage. Your TV is only 720p which means the resolution is probably only 1366 x 768 so you'll loose some coverage over the ASUS. It makes for an odd viewing experience.
 
I bought a 1920x1080 32" 1080p TV to use as a 2nd monitor but wasn't impressed with the screen quality so I don't use it for that anymore. To my eyes TVs as computer monitors aren't as good as a computer monitors.  Like you I also have an ASUS 24" HD monitor which looks much better than the 1080p TV. If it were me I'd go with an identical 24" monitor and leave the TV as a TV.
 
 
 
 
 

BTW MeshodMan,
 
Who's the general in this story?

Mesh
Nobody knows?
 
 

The question was too general. 

Both digital (there is an analog-y variant of the DVI connector, but let's not go there), you can just use a cable to go DVI->HDMI, so connect 'em up however you want.  Won't be a fat lot of difference either way though tbh.
 
Extend the desktop in the graphcs settings then you can set different resolutions for both screens.  In Windows 10 (from 8 onwards I believe), you can even set a different background picture, Wowzah Windows! ;-)

Seriously, Mesh?  A REAL question about computers in the Coffee Hause?  shouldn't this be moved to the Computer forum???? 

Beagle
Seriously, Mesh?  A REAL question about computers in the Coffee Hause?  shouldn't this be moved to the Computer forum???? 

 
Since it's just a "general" question AND there's a lot more "general" traffic in the CH, I made a "general's" decision to post it here . 
Of course, if you do feel it needs to be moved to the Computer Forum, I have no problem bruddah......(however, it does relate to more or less a "home theatre" (lo-fi) setting than an actual technical/Computer forum Pro setting no?) 

bapu
BTW MeshodMan,
 
Who's the general in this story?

Yo bro.....I.B. DUH general!!

was just a joke, mate.  

Beagle
was just a joke, mate.  

Ewe were jus yanking me chain?
Pulling me leg?
Taking the Mickey out on Goofy?
 
I so sowwy.......me just not sure if ewe were being optical or clownical.

Hi,
 
My preference and understanding goes with Bapu here ... you should have the same monitors connected to a video card that can support both ... and I am not sure that I would use a separate/secondary hook up for it ... I would recommend getting a better video card that can handle 2 monitors easily enough ... and yeah, I think that different resolutions will likely overwork the card a lot.
 
I have not regretted spending on my first "good" (or "great") video card, and spent about $300 dollars for one, but it has been worth it and then some ... (GTX 970 and GTX 760 and a spare GTX 780 that did not like one motherboard) ... and adding 2 monitors has not been an issue, since my mains are both DELL's and they are very similar in totality. 
 
I'm not the expert on this, however, and recommend a quick stop at the Transilvanian Brainfarming Factory and get their view first ... but in general, I do not think the video card can handle separate displays like that unless it is a duo type card, and those cost a lot, still ... and even I do not use those setups that use two cards on the same computer.
 
Hope this helps

The graphics card below is what I have and it does have 2 connections for a dual monitor setup (HDMI & DVI)....mine's just not the "Turbo" version, but everything else is the same.