Most people practice hands separately during the beginning learning stages of their piece. After they’re fairly comfortable with the notes, they practice hands together until they’ve finished learning the piece.
Although it’s helpful to practice this way early on in the learning stage, it’s often overused. Practicing hands separately is absolutely essential, but it only works well if it’s done at the right times.
So you look at your new piece of music, and you decide to divide it into sections. That’s great! You proceed to learn section one hands separate. Then section two, and three, and so on. You don’t always need to learn the whole piece this way. Sure, it’s great to break down the tricky parts and make them as easy as possible, but it often ends up being a waste of time if you practice the whole piece hands apart only for the first several days or hours of learning your new piece — especially if you’re just just trudging through it for the sake of getting your time in at the piano.
The most profitable time for hands separate practice is when you’re familiar with the notes and structure of the piece. Now it’s time to listen to what you’re playing (read Listening Vs. Hearing). Pay attention to chord progressions, decide on how you’re going to do your dynamics based on the harmonic structure. Practice extremely slow, and just enjoy the beauty of each chord. The piano is a beautiful instrument, but only if you take the time to bring the most out of every note.
The next time you practice your piece, play with just the left hand, and play really slow. Don’t rush through a single note. Let the music bring you on a journey. It may sound crazy, but you will understand what I’m talking about if you try it. So I hope you do! Let me know how it goes.