I think it's important to finish a song since it isn't a song until it's finished. Putting off finishing songs is a bad habit to get into.

This isn't to say that I never abandon ideas that lead me into a cul-de-sac but I think if I've got a verse and chorus it's important to finish it. I find that this helps me write better songs because I feel it refines my songwriting process and, like other posters have already said, it's worth while even if you only get one good line or a little thing that you can use later.

Although I'm getting on a bit I'm actually fairly new to songwriting and tend to have to get the whole of an idea recorded in one session before I lose interest or come up with something else, get distracted and move on.  I haven't yet gone back to a song and reworked it with any success which is why most of my stuff is based on very simple chord progressions.  I'm also not a very competent musician and get frustrated that I can't come up with the killer solo or kick ass drum groove for a track.

I've got massess of ideas scribbled down on bits of paper or recorded on my smartphone that have yet to make it into any structured song.

I also hate to listen to my own stuff when it's recorded, although I've been in a band for a few years now and am gradually improving my vocals.

If only I had the time and money to sit in a studio with a bunch of session musicians!! 

Bahaha I know exactly what you are saying man.

I usually have defferent phases of liking and not being sure of what I'm working on. But I've been getting better at finishing it anyway. I realize that the phases are created by my general mood.

When it comes to finishing songs, if I'm at all unsure, or stuck on the song, I make sure the finish it even if prematurely. That way, my brain can relax and either:

i) work on something new
ii) re-finish the song later on

Either way, if you finish your songs, you can relax about it and not have to worry about incomplete songs.