First of all, a premise: doing a full system install on a flash drive is generally not recommended, because it will definitely shorten its life. USB thumb drives are not suited for continuous, repeated writes on it and a full OS writes a lot of files while running.
It would be much better to use a live system with a persistence file. Now, about your problem:
EXT4-fs error (device sda1): ext4_find_entry:[number]
Your have input/output errors on the drive. This can have two causes:
- the file system is damaged
- the drive is damaged
In the first case, you can easily solve it by running a file system check. You can use the live USB you have (the one that works fine), run it and open Gparted after plugging the faulty drive. Then:
- select the faulty drive from the dropdown menu
- click on the main partition
- click on Partition → Check
- follow the instructions
However, if the drive is faulty there is little you can do. You should think about getting a new one before it's too late because you risk losing data.