They mean the same thing except that "난 당신을 사랑해요." sounds more dramatic than "사랑해".

난 당신을 사랑해요 is too formal. No one uses that expression, I've never heard of that expression in my whole life. So if you meet someone whom you really love, just say 사랑해(요). You don't need to say 난 because you are the one who speak it and 당신을, either, because your partner is the one who listen to your word. I know this is tricky but in common conversation between two, we'd prefer not to mention 난, 당신을(I and You). It's not necessary. for example.. A: Did you have a breakfast this morning? B: Yes, I did. you can translate it in Korean A: 너 오늘 아침에 아침 먹었니? B: 응, 나는 먹었어. But this kind of conversation is unusual and awkward. A: (너) 아침 먹었어? B: 응, (먹었어) it's much natural.... if you want to speak Korean more naturally, you'd better avoid to translate it.

난 당신을 사랑해요 is formal 난 당신을 사랑해요 is more complete than 사랑해요/사랑해 while 사랑해요 or just 사랑해 is informal.