Overdubbing is the process of recording different parts of the same song at different times. It would make absolutely no sense at all in a concert setting to have, say, Paul play his bassline to “Roll Over Beethoven” as a solo, then when he was done George could sing his lead vocals by himself, then when he done Ringo could play his drums alone, followed by John's rhythm guitar chords. It'd be the equivalent of eating a piece of cake but on the first bite you taste only flour, and on the next only eggs, and the next just sugar. Of course that wouldn't work – whether in music or baking, all of the ingredients must be combined to render the final product. But in the studio, you can do that because the tapes can be combined through overdubbing, the layering of these various components into the final product. While a few minor tracks were overdubbed on Please Please Me, on With The Beatles it was quite heavily. For example, the piano solo in “Not a Second Time” was overdubbed. The Beatles recorded the guitar, bass, and drums, leaving a space for a solo because they weren't sure what that solo would be. Overdubbing allowed the band to come back to the song later and add a solo, in this case played by George Martin.
Another recording tactic used frequently on With the Beatles was double-tracking, which is a particular kind of overdubbing in which the exact same thing is recorded twice, then layering on top of each other so they are heard simultaneously. The Beatles used this trick almost exclusively for lead vocals because the technique supplies reinforcement to whatever was double tracked, and lead vocals need to be strong. Obviously a singer can't sing his lines twice at the same time during a live show, but in the recording studio you can, and double-tracking is how that is accomplished. For example, Paul's singing on “All My Loving” is double tracked. You can tell because there are slight differences between takes – even though he's singing the same lyrics and notes. This is most noticeable on the word “I'll” as in “tomorrow I'll miss you”. The Beatles used double-tracking on 8 of the 14 tracks on With the Beatles ("It Won't Be Long", "All My Loving", "Don't Bother Me", "Little Child", "Please Mister Postman", "Roll Over Beethoven", "I Wanna Be Your Man", and "Not a Second Time")
and would continue to do so throughout the remainder of their recording career.
Additionally, the covers of the two albums showcase this change. Please Please Me's cover shows the four Beatles in a rather creative though quite "standard pop cover shot" pose.