A ball is thrown horizontally and at the same time another ball is dropped vertically from the top of a tower. Will both the balls strike the ground with same velocity?

No. The ball which is thrown horizontally from the top of a tower of height $H$ will attain a velocity of $v_R^2=v_H^2+v_V^2=v_H^2+2gH$

And, the ball which is simply dropped from the same height will attain the final velocity of $v^2=2gH$

Because of the factor $v_H,$ the ball thrown horizontally will strike the ground with more velocity.