They can both be used the same in 99% cases.
This is only describing a VERY SUBTLE difference:
"Give me a break" is maybe more to ask one person not to push so hard.
** Hey coach, we work hard, give us a break!
In this case the coach is pushing hard on the players. THey are not asking for a time-out, they are asking for him to "lighten up" a little.
"Cut me some slack" is used when the players in the above situation have some specific and legitimate reason for asking the coach to lighten up.
** Coach, this is my first time back on the field after breaking my leg, cut me some slack.
Here, the broken leg is the specific reason for the player to ask the coach to lighten up.
However, using either phrase in either situation above would not sound weird or be incorrect. They are basically interchangeable.