“Birlikte” and beraber” are both adverbs and indicate “togetherness.” (1) • Yeni bir hayata birlikte / beraber başladılar. [They started a new life together.] In this context they could be used interchangeably. On the other hand, in the following examples “beraber” seems to be the most contextually appropriate option: (2) “beraber” in this context could be perceived as “despite”: • Eskiden oyunlar çocukça olmakla beraber daha samimiydi. [Despite being childish, the games in the past were more sincere.] (3) In the sentence given below “beraber” means that there is no gap between the house and the ground • Evin merdiveni yok; ev toprakla beraber. [There is no stair (in front of the house); it is located at the level of the ground – it is directly grounded on the floor/soil.]

I agree with Yasemin. She explained very well, however, both are fine for second one. You can use them interchangeably. Eskiden oyunlar çocukça olmakla beraber/olmakla birlikte daha samimiydi.