As everyone knows, our own voice sounds different to ourselves. My question is if there is an actual pitch difference and if so, how much it is?
There is a preliminary question hidden here which is, "Does the timbre of a sound affect the apparent pitch of that sound?" The answer is yes. It took about 30 seconds to find a study that had this finding here (note this paper references at least one additional study showing a link between timbre and perceived pitch):
The conclusion drawn from the study is that timbre-induced pitch shifts may attain magnitudes that are likely to lead to conflicts between subjective and fundamental-frequency-based pitch assessments.
I have a slight objection to the choice of words in that quote, since my understanding and preference is that the word "pitch" always refers to a subjective sensation, that is largely based on the frequency spectrum of a sound, and not entirely on the lowest frequency.
In any case, timbre does affect pitch, and as you note, a significant portion of what we normally hear of our own singing voice has reached our hearing apparatus via very different path from any other listeners, so we experience a different timbre of our own voice from the audience.
This could affect our sense of our own singing pitch. My personal experience as a singer is that it does affect my own sense of the pitch of my voice, but only very slightly.
Since everyone's vocal and hearing apparata and intermediary tissue and bone are unique to them, we can't expect that everyone will experience any pitch "deception" (if you will) regarding their own voice, and if they do, we should expect the amount to vary from person to person. In any case, we should expect it to be very slight. As commenters have noted, people are able to sing in tune, so any "pitch deception" must be either too small to be noticeable or not so great that it is hard to learn to compensate for it rather quickly.
I think the downvotes to @Tim's answer are rather unfair. And @Todd Wilcox has taken a study that states 'something MAY happen' and escalated it to 'something WILL happen'. Yes, timbre CAN affect pitch perception. But perception is a very fluid thing. Many singers DO sing in tune. They have adjusted their perception of what they hear (which MAY involve a pitch-shift) to correspond with what the audience hears.
Anyone who has done manual pitch correction in AutoTune or similar programs will know that perceived pitch involves a lot more than the centre frequency.
To answer the original question. There MAY be a perceived pitch difference. Or there may not. If there is, good singers adjust their perception to compensate.
I have tried to help one particular singer who makes a very nice sound, consistently a semitone below the pitch of the music. Still struggling with that one!
The difference is in the timbre and volume but not the pitch. We would not sing in tune if that was the case!