The difference between beide and die zwei is quite subtle. Beide means that two persons, things or animals have certain characteristics. It refers to each person, thing or animal separately. Die zwei refers more to two persons, things or animals that are seen as one unit. But be careful! Die beiden refers again to two persons, things or animals in unit.

Some examples   Beide kaufen sich jetzt ein Auto. = Both buy a car. (Each of them now has a car.)   Die beiden kaufen sich jetzt ein Auto. = The both buy a car. (One car for the two of them.)   Mir gefallen beide. = I like both. (I like each of the two.)   Mir gefallen die zwei / die beiden. = I like the two of them. (Together I like them.)   Beide haben gelogen. = Both lied. (Each one of them told a lie.)   Die zwei / Die beiden haben gelogen. = The two lied. (The two together lied, not necessarily independently from each other.)   Beide haben ihn gefragt. = Both have asked him. (In total he was asked two times.)   Die zwei haben ihn gefragt. = The two asked him. (In total he was asked just once.)

Mixing beide with die zwei / die beiden does usually change the meaning of the phrase only a tiny little bit. However, sometimes it can sound a bit weird die zwei/ die beiden instead of beide.

Examples     incorrecto: Ich habe mir die zwei Arme gebrochen. = I have broken the two arms.   correcto: Ich habe mir beide Arme gebrochen. = I have broken both arms.

In English there is the same difference. Therefore it should go right without thinking about the whole topic too much.