HIVE tokens can be staked (aka "Powered Up") in order to have HIVE POWER. Therefore a simplified explanation is Hive Power is Hive set aside to give benefits on the Hive blockchain.
You get to do more transactions on the Hive blockchain.
Transactions like comments, posts, transfers on apps like PeakD that focus on the blogging aspects of Hive. But also note there are many other types of application on Hive... so more Hive Power may let you play more games on a Hive based game like Splinterlands.
Your vote on the Hive blogging system is worth more.
When you vote on posts on sites like PeakD.com your vote is more valuable the more Hive you have staked into Hive Power. A vote is like you saying you have more shares of a stock and you're voting where you think a portion of that days INTEREST should be allotted. At the end of 7 days every voters votes are accounted for and at the end of 7 days a vote reciever will get rewards based on all those votes.
You get to vote on governance of the Blockchain.
Hive is not owned by any one person or group or company and decisions of the blockchain are made by a type of Board of Directors and every single hive owner gets to vote for those directors (called witnesses) and the vote is proportional to the Hive Power you have.
You get to vote on projects that get funding from Hive.
Similar to voting for governance a project/person can ask for funding to do something and people vote if they would like to see it get funding.
You can delegate Hive Power to help others
So if others need hive power to make more transactions you could help them out.
Hive Power also earns a small amount of interest that is earned just by holding it.
So what's the benefit of voting on content
One benefit is that votes on content earn the voter some of the rewards... so the larger your vote it proportionally makes your reward for voting larger.
Another benefit of a vote on content is you help in the sorting/ranking of that content. Whether it's 7 day ranking inside a topic page or in some community pages. Or positioning of comments inside each post. You can basically be a part of curating what people may see on a page or a post.
However with just a few HP (hive power) you could make lots of comments. To constantly play a game like Splinterlands you may need more in the 20-30hp range but that's to play a LOT.
Other than that there are very few NEEDS ... mostly it's just wants if you want to have a large vote for posts or vote in the governance.
Hive tokens are liquid... meaning at any moment you can transfer them either to another user or to a crypto-currency exchange. And in currency exchanges you can for example find a path to a fiat currency like US dollars. (Hive Power on the other hand can not be transfered it can only be powered down in a long process back to hive tokens)
Hive Tokens don't have any additional benefits... that's basically the biggest difference. You can transfer them, pay for stuff, exchange for other currencies like Bitcoin. But there are no unique benefits from the blockchain for holding a lot of them.
@peakd. Main authors presently: 
@jarvie