Is there a difference between 4 4 time and common time in music?

For any practical intents and purposes, there is no difference. The "C" designator means exactly the same as 4/4.

https://en.wikipedia.org/wiki/Time_signature

Why are there two symbols? The 'C' time signature is a relic from the very earliest days of music notation. The use of the symbol is curiously tied to the Christian Holy Trinity where 3 beats to a bar was considered perfect (because that related to the 'Trinity'). That timing was represented by a full circle. Music composed to any other time was considered imperfect and was represented by various incomplete circles. The half circle represented 4/4 timing and looks like a C.

The sign that can be used instead of the usual 4/4 time signature looks like a capital C. A lot of people think that it really is a C, and that it stands for "common time", but that's not really very accurate. [...] ...music with 2 or 4 beats in the bar was considered "imperfect", and was represented by a semi-circle - which happens to also look a lot like a C!

https://mymusictheory.com/more-music-theory-topics/time-signatures-4-4-or-c/