Lt. Bob made a pretty good observation but you'd really only notice the quarter step if there was a background vocal along with the song singing harmonies.
The guitar part is fairly stagnant so you don't get a lot of colliding pitches with it, and its just a melody line so a quarter step off would very easily sound like a note that just barely sags rather than causing a majorly discordant sound wave.
A quarter step is such a small interval that a vast majority of people (even a semi-trained ear) won't notice unless there is a competing or complementing tone for it to be compared to.
You would think that singing legato would make singing individual notes easier but thats not necessarily the case. They involve two types of athletic abilities in terms of breathing and controling your airflow. This is true much the same way that singing 8th or 16th notes doesn't make singing a whole note easier, is probably makes it more difficult (because you have to change your entire counting frame of mind.)
I also have a new and much more advanced practice technique for you if you want it. Seeing as how we're talking about quarter steps and such this fits in well with the discussion.
Choose some time frame for your self. 4 beats, 8 beats, 16 beats, etc (the longer the time frame the more difficult) and then choose a tone/note. Slide down or up from the tone/note that you chose (say Middle C) to either B or C# within the amoutn of beats selected. You should hit the target pitch on the last one or two beats.
The trick is to not hit C# with 6 beats left out of 12, but to hit all the middle tones between there, if you can. At first this sounds easy, but it is actually extremely difficult and requires a great deal of breathing skill and also pitch sensitivity. I would highly suggest you not try this until you are well versed with the other exercises listed up above.
What mic and pre are you using? Please tell about the room and mic positioning too.