Well duh, I can almost hear you saying. Of course, they are different: one deals with paper money of which there is an infinite supply, the other with your own money, which unfortunately is limited.
The other big difference, not so obvious for someone who just starts out in this business, is the trading performance. Almost everyone has witnessed very good results in a paper account which encouraged them to open a real account, where the performance would suddenly become lackluster at best or complete failure at worst. Why does this happen and what can be done to solve the problem?
Before I start, I’ll assume that you know what you’re doing and you have a trading strategy. If you just pushed some buttons and got lucky, that’s a different story. It should go without saying that your luck will run out sooner or later (usually as soon as you open the real account 
). I’ll also assume you had the demo account with the same broker where you opened your live one, and you used roughly the same amount of money (trading demo with a $50k account and live with a $500 one will just make your life more complicated).
OK, now with that out-of-the-way, I believe the most important reasons for the difference in performance are your emotions: your hopes, your fears, your greed.
The speculator’s chief enemies are always boring from within. It is inseparable from human nature to hope and to fear. In speculation when the market goes against you, you hope that every day will be the last day and you lose more than you should had you not listened to hope to the same ally that is so potent a success-bringer to empire builders and pioneers, big and little. And when the market goes your way you become fearful that the next day will take away your profit, and you get out too soon. Fear keeps you from making as much money as you ought to. – Jesse Livermore
You see, while paper trading, there were no emotions because no real money was involved. But as soon as you have your hard-earned cash on the line, your emotions start wreaking havoc with your strategy. For example, out of fear, you’ll close trades before they reach the stop-loss, or you’ll ignore it hoping that the market will come back. Out of greed, you’ll trade larger, hoping to make money faster.
The solution would be to open up a real account just with money that you can afford to lose, as the omnipresent risk warnings state. In other words, with risk capital that you have no emotional attachment to. It won’t be exactly the same as with a paper account, but it’s as close as it can get. Remember though, for all this to work, you first have to develop a sound strategy that fits your personality. Eliminating emotions alone won’t transform a poorly-conceived, losing strategy into a winning one.