The tour that I booked already has an entrance ticket included. Are
there separate tickets for the underground area? If I book the
underground tour, is it essentially the same information as the
walking tour?
Julie, while your tour includes entry, you can't use that same ticket for a second entry to do the underground if that section isn't included on the tour you've booked. You'll have to purchase another one as your underground tour will be a separate booking at a separate time. I've no idea if the information on your walking tour will be the same as for the underground or not as I've no idea what walking tour you're taking (and haven't personally felt the need for walking tours in Rome so far).
Also the website you provided, is it only valid until Dec 2016? I am
planning on going to Rome in June 2017.
Right. As stated above, you can only book underground tours through that site on every third Monday of the month for the following month. So June bookings aren't available until the third Monday in May at 9:00 Rome time. These tours fill so rapidly for high season that previous posters here have said that they've gotten up in the middle of the night to book. 9:00 AM in Rome, for instance, is 2:00 AM where I live.
You may also purchase your tour and entry tickets by phone if not wanting to do it through the website but again. you may very well be doing that in the middle of the night to nail a spot.
Individuals: +39 06 399 67 700
Monday-Friday 9-13:00 and 14:00-17:00, Saturday 9:00-14:00
I don't know who you booked your walking tour through but another option is to see if they offer one which includes the underground (if not included in your current tour itinerary) and if so, switch your reservation if possible.