This isn't going to be anything really flashy. I'm not trying to explain the intricacies of the new edition to you yet because I don't know them and neither does anyone else. Anyone who makes predictions about most anything regarding 6E right now is an effing idiot, you understand? We still haven't even seen the whole rulebook yet, We haven't seen the FAQs. We don't know how this is going to go.
That said, here are how I think a couple of factors (out of hundreds) are going to influence the game. The first part of seeing the whole is seeing the parts.
The change to 5+ cover saves being the standard is a pretty big deal, because 4+ cover in the previous edition was a huge part of how 5E changed things. Not ALL cover is going to 5+; it seems that buildings, ruins, etc, will still be 4+s, but most things will not be. This drives things greatly away from the torrent-of-fire standard that was 5E and it makes armies that have weak armor saves (notably IG, Orks, and Tyranids) a lot weaker in some respects. MEQs gain a lot of ground with this as you need more specialized weapons to kill them, and you WILL be seeing those guns on the field more often now, especially since the absolute prevalence of vehicles will ebb away somewhat.
Looked at numerically, 5+ cover may not seem like a big change because, hey, that's the smallest change that can be made, right? But think about it this way- you're failing 33% more saves (2/3 instead of 1/2 of the saves you roll.) Flipped the other way around, going to 4+ cover is a 50% survivability boost, so the change is VERY significant. When shooting happens, models are going to die more now, and it's as simple as that.
Rapid Fire also sees a very significant shift, this time for the first instance since... uh, the ending of 2E, I think. Rapid Fire has consistently been something that limited your ability to fire on the move and that made the game very static; your mans mostly sat still and did their thing. If they were able to move and shoot, that was a big deal. But now you can move and fire those Rapid Fire guns to full distance, which closes the gap between them and Assault weapons. A Bolter still isn't a great gun and you'd really rather have... well, almost anything else, but neither is it a real hindrance, as you will often be able to toss out some shots to put wounds on enemy mans.
This change to Rapid Fire is one of several in the book that seem to be pushing the game towards a more mobile, dynamic feel; 6E feels like it's going to involve a lot more guys shuffling around, though whether it will be of particular import or just "I move six you move six no effect" is hard to say. But the potential is at least there, so that's something.
Snap Shot is another big addition- basically, most of the times when you wouldn't get to shoot a weapon before (extra guns on a moving vehicle, moving a Heavy weapon, etc) you can still Snap Shot, and even a few unique times like the charge reaction. Snap Shots always occur at BS1 and you can't Snap Shot a gun that doesn't use Ballistic Skill (although it's also been said that templates get d3 hits during a charge reaction, so this may not be entirely correct.)
Snap Shot is more beneficial to armies with low BS scores, as they generally have more shots to make up for it and are less penalized by the reduction. Orks, of course, are the kings of this- it halves their chances to hit, but they tend to be tossing out so many shots anyways that it hardly matters- charging a mob of Shoota Boyz can be a very dangerous proposition now. Guard, Tau, and other BS3 armies also fare relatively well, especially since most of them are shooting armies anyways and any chance to get extra shots is a good chance. Many players pooh-pooh it, saying that the sixes to hit mean it won't matter, but the Necron Rule applies here- sometimes you will just roll sixes and it will be awesome. Don't count on getting it, but on occasion your Plasmagunner or Sniper guy will totally save the unit's ass by rolling like a madman and crippling the enemy assault.
Snap Fire is also needed to hit any flyer in Zoom mode (oh god I already hate typing that, I feel like a preschooler; more so than normal, I mean) as well as certain other circumstances, such as the aforementioned flyer shooting at non-ground targets. The interaction with flyers will depend a lot on what the FAQs tell us, but if there is not an abundance of good anti-flyer weapons for all armies they could quickly become very unbalanced. Expect to see many, many different "flying circus" armies that try to overwhelm you with aerial targets and reserve manipulation.