Both of these are top grade Deer cartridges & would give you similar results when looking at performance alone. You asked two questions.
(1) Which has less recoil? the 25-06 as you are using the 30-06 case for
both & normally heavier bullets for the .270.
(2) Which is more accurate? This is more of a factor of barrel quality, bedding, etc. & you will have more variation from gun to gun whether you had a group of 270's or a group of 25-06's. Their potential is very close.
This is how I would answer this question in a general sense if the type
of gun was not discussed & we are talking about the cartridges alone.
However, this is the Handi section & you mentioned your Handi, so I am
assuming that you are in reference to these rifles. If so consider this, the
25-06 has a 26" barrel & the 270 has a 22" & maybe a 24" is also available. If you handload, this will enable you to get max velocity from the 25-06. If you look at all factory "Deer" offerings for both, the .270 Light Mag. may give you a little flatter trajectory, but with the best loads for both the 25-06 will be as flat or a hair flatter. You can also work up better Varmit loads or if you shoot factory loads, you can get the 85 Grain Bal ST which is great. The barrel length, contour & balance is just better in the 25-06 Handi for longer shots.
Also, because of the difference in length & weight of the 25-06 & 270 Handi barrels, the 25-06 recoil is reduced even further. Yor can get a
reduced recoil load now for the 270, but a 100 Gr. 25-06 load is already
a reduced recoil load.
As far as number of loads available are concerned, the .270 has more but
there more than enough for the 25-06 in factory loads & a huge selection
of bullets for reloading. Concerning penetration, the smaller the caliber,
the more you should judge bullets wisely. This is easy now, as there are
so many great bullets available. I have never recovered a 100 or 115 Gr.
Barnes TTX or a 120 Gr. Nos. Part. from a Deer & they always penetrated
enough for Hogs. I have heard the same about the 115 Nos. Partition.
One thing to also consider is the size of your son & if he has to hold the
rifle offhand at times or will he always have a rest. I say this because if
he is big for 13 he will be able to hold the heavier gun in a shooting position fine, but if he is small, you may need to be sure. My youngest Son started using the 25-06 Ultra at about 12 & he had no problems at all.
I guess it is evident what my choice would be. You can always get him
a bigger cal. barrel when he is ready for Elk size game.
I apologize for trying to build you a watch when you asked what time it is,
sometimes I get carried away!