That is not to say that neither can be done the other way; the main contact a PVE player will have with PVPers is pirates can flipping or baiting, which is usually a solo activity, at least until the pirate hooks a PVE fish. Pirates can also often be found solo in low sec preying upon new players who wander in acidentally and do not yet understand the game mechanics, or other pirates in lesser ships.
However the bulk of PVP is conducted in fleets with a very different mindset. The differences are:
PVE missions and belt ratting provide a steady income in usually quite small increments. PVP fleet warfare often means long periods with no, or even negative, income punctuated by occasional massive payoffs. Usually those payoffs are from PVE goals (officer spawns in lowsec complexes that require running the gauntlet of other PVP fleets, for instance) or from killing those foolish enough to fly in lowsec with hideously rare modules.
2) Ship Loss.
A PVE player will only lose a ship if they make a mistake, on average once every few months or so. A PVP player has to leave station with the mindset that they will lose the ship more often than not. This is where the necessity for fleet warfare comes in. In PVE you can safely fly an expensive ship because you know exactly what you will be facing. PVP is almost always an unknown quantity where anything could happen. It is better to have three ships that combined have the damage of one when the cost of the single ships is three million each and the single ship is 70 million.
This is the one that the PVE player gets hung up on the most. I have spoken to a lot of players who have said to me that they are waiting till they get better at the game before attempting PVP. I have to admit I have been guilty of this in the past myself. However it is important to realise this: PVP DOES NOT REQUIRE MORE SKILLS THAN PVE.
Keep repeating that to yourself until you understand it. Different, maybe. More, no. Any holes in the quantity of SP you have can usually be made up simply by having a larger fleet.
There are some skills useful in PVP that are largely useless in PVE for some members of the fleet:
MWD instead of Afterburner
Recon Probing skillset
Electronic warfare skillsets (sensor dampening, sensor jamming, webbing, etc.)
Remote projection skillsets (transferring shields/armor/cap to other members of your fleet)
Cap draining skills (though these are less useful than they used to be)
Also in General guns work better than missiles, which is not to say that missiles are useless.
Just as in PVE of course more skills are better than fewer but don't think you need many millions of SP before you can join in. There will always be someone bigger and better than you but that shouldn't put you off having a go. A brand new character with the right skillsets can fly an electronic warfare specialist ship with remote projection and ew modules all ready to use, and has the advantage of it costing nothing to replace a clone should you get podded.
Until now PVP was fairly random and involved a lot of waiting around and taunting before it would begin, whilst PVE had missions which could be logged into anytime you felt like shooting at something.
Now with the advent of Factional Warfare there are specific goals that can be completed anytime you wish, even if there is no other PVP player around. Like a PVE player you can log in anytime, get a mission, and set out to complete it. There will be some NPC players to fight, as in PVE, but there may or may not also be PVP players there to try and stop you too. Missions will change each time and not be set and knowable in advance. They will probably have a fleet, but there may be one or two or even no other PVP ships there at all.
This is Factional warfare. This is the Empyrean Age. This is now. Get in your ships and take part.