First, a caveat - the information below should be taken, as any kind of business writing, with a grain of salt. Take from it what can be validated by your experience and acumen, use your head and say no to formulaic processes.
 
Positioning vs Differentiation
 
Differentiation is to positioning what bad attitude is to toddlers. You can't have one without the other unless something is terribly wrong. Though often talked about in the same books at the same time and being as tightly intertwined as they are, positioning and differentiation are two different beasts. The former deals with what consumers perceive you to be, whereas the latter has to to with what sets you apart.
 
Differentiation doesn't depend on the end-consumer. Positioning does, and is often influenced by it. A common formula for writing a positioning statement goes like this: TA + FOR + POP + POD + RTB. In plain English, you have Target Audience, Frame Of Reference (or category), Point of Parity (what defines the category, what everybody's product does), Point of Difference (what makes you unique) and the Reason To Believe (the rationale behind your POD).
 
Without a POD, you're a generic, me-too brand. You don't shine in any way, but you're not bad either. You're not the cheapest, nor the most expensive. Not the best, but not the worst. You're mediocre. And who wants that?
 
Gaining ground
 
In a crowded category, if you're not one of the top 3 players, you'd better find a niche audience to appeal and market to. It can be rather expensive to try to keep up with the big guys, especially if you're short on media money. The more money you have, the more media you can buy in bulk and the cheaper it gets due to volume discounts. In any case - if you're #2 or #3, you can do at least 3 things:
 
1) Tackle the leader and try to demote its key differentiators/ make them seem like not important or not relevant
2) Tackle the leader and make their differentiators your own (aka remove their POD)
3) Own your position and wear it as a badge (see the oft-cited Avis example
 
However, your biggest problem will be getting a high share of voice. This means getting as much coverage (paid or not) as possible. In a crowded market, brands tend to become commoditized, so their PODs have less to do with actual features and more with lineage, brand likeability and exposure. This can be both a blessing and a curse, depending on the age of the category - it's hard to think that Coca Cola and Pepsi will fall from their positions anytime soon.
 
I'm my own category
 
If the headline above applies to you, you're either a lucky visionary or naive. But let's say you are among the first to create a new category. Your best bet is to own it and make people associate you with the category. Think Xerox. Think Google. If you can become a household name, do it. Share of voice becomes share of market.
 
This usually applies when you're not the only one in a category, but one of the small and few players squirming for users. However, if you're truly the first, you can do something else to gain traction - steal from indirect competitors. Margarine did it with butter. Apple did it with IBM. Uber is doing it with taxis. Coca Cola can do it with freshly squeezed orange juice. As long as you solve the same need, you can attack them, provided that your differentiator is strong enough. What makes a differentiator strong, you ask? 3 things:
 
1) It's technically complicated (or expensive) to reproduce ; in "0 to 1", Peter Thiel tells us that you should launch only if you're 10 years ahead of the competition, technology wise. That's a bit extreme, but 1-2 years should be enough to allow you to grow before others do.
2) It takes time to replicate (think infrastructure)
3) It's relevant to the consumer - without this, the points above are 100% worthless.
 
To sum up - if people can choose between you and some other solution, give them a relevant reason for which they should choose you. That's really all there is to it.