Will this run crysis on max settings?

no it will run on high ish settings but not max the 6670 is a low end card you need a mid range card to max out crysis and get 30 fps minimum. your system is fine for it on the whole just a little weak on the gfx. for 650 its a very nice build. maybe in a few months you can upgrade that gfx and move from low mid to high end. a small tweak here and there and you will have a very nice pc.
also your hdd is poor. you will get a lot of texture pop because its slooooow much better to get a 7200rpm 1 or get 2 and run them in raid 0.
even if you get 2 500gig 1s you will get much smoother gameplay and system experience.
i run 2 of my 3 1tb spinpoint f3's and get an average of 180MBps read and write. which is over double what yours will top out at... it really is a good investment if you dont want to get a ssd that costs way to much for the storage you get...
(reason i say this is some manufacturers have just cut production because ssd prices were coming down to fast, they basically are holding the price up which is unfair to the end user)

My 6970 struggles with Crysis maxed out sometimes. I would say medium settings at best.

Crysis runs like crap on every machine!

Runs great on maxed on my, i5 1500k and gygabyte gtx 570 super overclocked.

My 6870 runs it maxed out at 1080p no problems with framerates at all.
A 6670 will run Crysis but definitely not maxed out

+1 for 6870 running crysis maxed. 6670 on colleagues machine won't cut it especially the big ending.

Q

Ah forgot t mention, I did get a lot of lag in the final cutaway scene with the massive whirlpool but part of that might have been drivers for W8 not being quite ready. But it's not actual gameplay so who cares