I'm just wandering if there is any advantage to neck sizing the 375 H&H?

Are there any dissadvantages in doing this?

I am currently full-length resizing by brass and have so-far reloaded them 6 times, and are still in good condition.

Would neck sizing increase the case life of a calibre like the 375?

Also, can anyone tell me how many times a 375 H&H case can be safely reloaded?

Whew!,
Bit of a can of worms, but here goes:
Brass life will depend on load pressures and how hard the brass gets worked in the reloading cycle. Get a big paper clip and straighten it mostly out except for a little 90 degree hook at the end. Use that to feel for any thinning of the case wall just ahead of the web. As soon as you feel some it's time to toss that batch of brass.
How much does your brass change in dimension when you fire it? If your chamber is loose you might consider neck sizing (with the usual proviso that any brass for hunting be relatively fresh, and be full sized).
Another indicator of brass life is how often you have to trim it. I usually toss magnum brass when it needs to be trimmed for the fourth time.
Even if your brass is not thinning or flowing it will become work hardened after several reloading cycles. Then it takes a bit of a "set" in the fired state and springs back when you run it through the sizing die. At that time it needs to be annealed. I do anneal my brass after 4 or 5 reloading cycles. It is a bit of a pain, but nothing to write home about. The annealing softens the brass at the case mouth so that you can adequately size it again.
To answer your question more specifically:
If your chamber is not too loose, and your loads are on the mild side, you should be able to get 7 - 10 reloading cycles with one annealing in the middle. As soon as you notice a brass death sign in any case (thinned body, time for fourth trimming, split neck, loose primer pocket, etc) it is time to pitch the whole batch of cases.
As a side note, with my 416 Rigby (a low pressure case) I can get 15 reloading cycles if I treat the brass right.
Theoretically slope shouldered rounds like the 375 H&H and 300 H&H flow faster than sharp shouldered rounds like the 223 WSSM or 284 Win. But, every gun has its own story to tell you, so pay more attention to what actually happens than to theory.
Good shooting,
JCN

Probably the dumbest thing anyone can do is neck size any dangerous game rifle if he intends to hunt dangerous game..every year we see some pinhead trying to stuff one in the chamber that won't go, including yours truly on one ocassion

Today I use only new resized brass for dangerous game hunting...

that said, for shooting purposes it works quite well to neck size or at least size the belted cases on the shoulder of your chamber to an ever so slightly less than crush fit.