I always thought that a molecule couldn't be chiral if it had a plane of symmetry. Doesn't the chiral molecule here have symmetry diagonally across the OHs? The achiral one doesn't look symmetrical to me, so I also don't know how it is achiral.
Regarding the first molecule: I understand that the simplistic Fishcher projection - with all its bond depicted by solid lines - can be very confusing at first glance in determining optical activity. You may believe it to be optically inactive, hoping for a center of symmetry to exist, but it is not so.
You must keep in mind how the molecule actually exists in 3d! Recall the rules of Fisher projection, the horizontal bonds project outward, while the vertical bonds extend inward. Hence, the molecule would actually look like this:
Now, I hope you can tell, there is neither any center of symmetry nor any plane of symmetry here. Hence, the molecule is chiral.
PS: Your second molecule is chiral by similar logic, as also mentioned in the comments, and as you yourself noted.