The TA is the more classic tour and has a very wide variety of topography, i.e. seascape in western Oregon, then lush forests, then arid scrub land in eastern Oregon followed by some more less lush forest in Idaho before reaching Kooskia, ID where the TA and L&C merge. The TA is probably more rural and less traffic percentage wise than the L&C.
The L&C is probably more arid overall and has more traffic but does have some beautiful spots and pedistrian/cyclist only sections. The L&C is probably a little flatter percentage wise than the TA and it is obviously much shorter.
I would suggest since this is your first tour, I would suggest the TA since it is THE classic tour. As Babe mentioned, Oregon is not cool and comfy in August, at least east of the mountain range. However, the TA is cooler.
To give an idea the following are the 30-average highs (rounded to nearest whole number) on August 15 for various spots along the TA route. In Oregon: Astoria-69, Salem-83, Eugene-85, Sisters-82, John day-87, Halfway-87 (was 100+ in June when I rode it up that really long climb). In Idaho, Cambridge-90, Grangeville-83. For the L&C the temps are Portland-81, Hood River-86, The Dalles-88, Wallawala-85, and Lewiston ID tops out at 90 as you have a massive climb up onto the plateau (beautiful climb though). These are averages so the actual temp could easily vary by 10* or more.
I would suggest you go to CrazyGuyonaBike.com and review some journals from both routes. I would also suggest you look at Google Map's streetview along various points of the route so you can see the scenery. Between that and the above, you can make a decision on what is best for you.
I forewarn you though, if you go on a tour, you may never stop! Have a great ride, John