Dost is from Persian and Arkadaş is Turkish. Literal translation is the same. But that word has been in Turkish even before the 13th century. In contemporary Turkish its meaning has become a little different.
Acquaintance > Tanıdık
Friend > Arkadaş
Close Friend > Yakın Arkadaş
Bro > Kanka
In terms of closeness Dost is on par with Bro. But bro is too perky. And Dost has a greater meaning than close friend.
Dost isn't your party friend (this doesn't mean you can't go holiday or have funny things together). There is a level of seriousness and mutual respect but also intimacy. In your hardest times you would tell your secrets, problems to your Dost. You would engage deep personal or philosophical or political (etc) conversations with your dost. Your dost would support emotionally in big events of life like wedding, kids. After marrying dosts dont lose contact with each other and visit each others home with their wives/husbands. Aside your family members and your Wife/Husband, dost is/are your only other lifelong companion.
In Turkish close friend doesn't fit to all of this.