Both are translated into ‘with’, but there is a difference between the two. 

I think ‘bersama’ is used when you want to describe two items (or more) being together, or will be together, in one place. So when you want to say “I am going to the party with her”; you can use the word ‘bersama’ in this sentence: “Aku mau pergi ke pesta bersama dia”.

The word ‘dengan’ can also be used in the sentence above and can also have the same meaning: “Aku mau pergi ke pesta dengan dia”. But ‘dengan’ has another meaning too. 

1. When you want to describe what tool you use to do something, “She cuts her hair with scissors”; in this sentence you should use ‘dengan’: “Dia memotong rambutnya dengan gunting”. 
2. Or when you want to describe a manner or attitude of someone when doing something, “You need to handle it with care”; here you also should use ‘dengan’: “Kamu harus memperlakukannya dengan hati-hati”. 
3. Or when indicating separation or removal from something, “for a mother, to part with her child is one of the hardest things”; this also should use ‘dengan’: “bagi seorang ibu, berpisah dengan anaknya adalah hal tersulit”.