Doesn't it use the interface clock?

Good point Geoff. I would think that too. The Interfaces Audio driver is in control of the word clock for sure. I guess the question is, does the word clock control the DAW's tempo? If this is the case then one would think that there will be no drift if you use the same interface. 
 
The other question -- if we switch DAW's but are using the same interface and the same computer if there is a difference? If so then it is the DAW that crunches the numbers.   
 
I use Jeff's trick when dealing with re doing old Stereo backing tracks.
I have the MIDI file, and I have a Stereo Wave file that is a recording of both the original MIDI playing a hardware synth and real bass. 
So I know the original tempo buy opening the midi file.. 
When I drop the audio file into the project there is always a drift and the tempo needs a  tweak of a  +/- .02 or so to make the transients line up still at the end of the track. 
You'll also hear a flanging effect when drums are not perfect. 

Your DAW can use a variety of clocks. For example, SONAR has a dialog for choosing which to use for the Audio timing. DAW MIDI generally uses a different clock source and there can be several sources with the driver using one of them.
 
 

Lots of great information in this thread. I like the metronome idea too. 
 
As to different clocks, I think most clocks these days are going to trace back to a crystal oscillator, so you will have associated accuracy issues. I know there are ways to improve accuracy, like keeping crystals at a constant temperature, but don't know how widespread that is in the real world.

IIRC, the audio (word) clock and the MIDI clock (DAW/Tempo) are freewheeling with respect to one another.  There are usually plenty of 'sync points' in an arrangement, i.e. wherever the DAW starts playing a new clip of audio.  If you had an enormously long single audio clip, and an equally long single MIDI clip, chances are they'd start to drift with respect to one another after a (long) while.

I too like the idea of recording a click as a reference point. It just makes sense.
 
Just now I did record a click at 100 bpm for 5 minutes and exported it to Sony Vegas including a SR conversion from 44.1 to 48, to my eyes and ears it lined up just fine.
 
Very good points about longer bits drifting and differing interfaces having some affect.

In the start DAW the internal grid, the metronome and the music are all in sync. If you can at least migrate the music and the metronome to the second DAW all you have to do there is adjust its grid and you should be able to get all three lined up pretty well.
 
The only thing to mention here is remember to turn the audio metronome track off again in the second DAW otherwise you may hear two metronomes. Not a bad thing but may cause confusion.
 
Lately I have been converting the metronome to audio myself and actually editing the metronome track audio.
 
For example in these tracks that I worked with Starise on, the first one was the remix so the drums are at normal time. But the Cinematic mix has the drums at half time. I edited the metronome audio so the metronome was clicking at half the speed. It felt better playing the drums to the half time metronome track instead.
 
https://soundcloud.com/jeff-evans
 
Listen to Cinematic Mix 5. I have just done this and it my best mix yet.
 
 Here is a better quality version of the cinematic mix:
 

https://www.hightail.com/download/UlRRblRpOC9UWURsZThUQw
 
Another nice trick too is to move the metronome track so it clicks on the off beats. eg the 'and' in between the crotchets. That way your on beat playing is always against silence from the metronome. That feels very good but it does take some skill to not get back in time with the metronome on the on beats.
 
 
 
 

I've imported an amount of audio recorded elsewhere, some in Sonar, some not and most all given the BPM to me for reference.  I've never had another Sonar audio file not line up in my Sonar sessions.  It's always worked perfect.  I've had not the same luck with other files and typically will start to drift after a minute or two.  Usually a little slice, using my ear as a guide, always has worked fine enough.
Also, every instance of MIDI recorded and played back on a hardware piece, such as a keyboard sequence (Aka Roland, Yamaha, Korg, etc), they are the worst culprits for matching tempos.  Lots of fractions to the .01 resotuion, to line up the MIDI in those case  (IOW, Roland keyboard MIDI at 100BPM  = 100.13BPM in Sonar, as an hypothetical example). 
MIDI not a problem obviously with adjusting tempos, but my preference is usually slicing and dicing audio, versus time stretching.

I have always found midi to be very good and fit the new project tempo very well.
 
I always try time stretching as the first option to get the whole track to line up with only a simple tempo change which by the way is easier in some programs.  The reason is that if you start cutting audio and you are not vigilant with how you place the starts of those individual audio clips then you have ruined a great performance overall with bad editing and placement.  I like to solo these things and listen with very serious intent as to how this clip is grooving with the project metronome.  And all the way through.  You may have to cut that audio clip into 3 more pieces even within a phrase to get it right.
 
If you time stretch the whole track so that the last 4 bars are in perfect sync everything from there back to the start will be perfect sync and the nuance of the total live performance will not be lost.  Less time later on too with editing because you dont have to waste any time moving parts around to sync to the metronome. The whole part is in sync because of the great playing of the musician at the start.  (Some music should not be edited timing wise)
 
But I also agree with mixmkr that sometimes you have to resort to cutting things up and moving them around.  I think in those situations you just have to be vigilant as to how parts are playing with the click.  It is easy to not hear them so well and let them go in the background of a busy mix,  not realising why the time of the overall music sounds a little sloppy.  You tighten up even one part and also move it slightly ahead too for some lovely pushing the time or groove there.  Suddenly the feel of the music tightens up big time.
 
In that music 'Zaphon' I spent a lot of time listening to individual parts in solo against the click.  Starise gave me multitrack sessions and I had to very much cut and fit things around.  Even though we both knew what the tempo was.  Also Starise started this at 120 BPM and I slowed it down to 118 because I felt that music sounded better at 118.  I also ended up sliding some tracks back a little early too like 8 mS because of the nature of the attack of some of his synth sounds. The parts sounded better a little early.
 
I can time stretch each individual audio clip over its duration as well and that is very cool for trimming up the ending of say an arpeggiated line in reference to the very start of it.  Tim did some parts later and gave them to me.  They really added some drama too.  Some of those things got out of sync and I ended up time stretching over an audio clip a few times.  Tim may have recorded the new parts at 120 and I had to adjust them slightly.  But that is different from what the OP was about.  Things certainly do work a lot better when there are no tempo changes involved.
 
In some ways it is also a form of art and expression as to how you allow any part to play against the click and over its duration.  Who says it has to remain as played.  It can be chopped and manipulated now in a million ways.

John
That sounds like a sample rate mismatch Alex not a tempo problem. 

Nope this a a professional producer that has had more hits than I've had sliced bread and a good technical brain.
Not as simple or as obvious as that. We are talking drift.