What is the difference between working for God versus serving God?

I don’t think there is any difference.

2 Cor 5:20-2 Cor 6:2
Therefore, we are ambassadors for Christ, God making his appeal through us. We implore you on behalf of Christ, be reconciled to God.  (21)  For our sake he made him to be sin who knew no sin, so that in him we might become the righteousness of God.    (6:1) Working together with him, then, we appeal to you not to receive the grace of God in vain.  (2)  For he says, “In a favorable time I listened to you, and in a day of salvation I have helped you.” Behold, now is the favorable time; behold, now is the day of salvation.

The above passage spells out that we should be working together with God, as does the passage below.

1 Cor 3:7-10
So neither he who plants nor he who waters is anything, but only God who gives the growth.  (8)  He who plants and he who waters are one, and each will receive his wages according to his labor.  (9)  For we are God’s fellow workers. You are God’s field, God’s building.  (10)  According to the grace of God given to me, like a skilled master builder I laid a foundation, and someone else is building upon it. Let each one take care how he builds upon it.

Heb 6:10
For God is not unjust so as to overlook your work and the love that you have shown for his name in serving the saints, as you still do.

So believers are working in God’s name by serving the saints.