So you are saying that you can not boot the live USB? Where does it hang?
Edit: You should be aware that selecting Try Ubuntu MATE without installing actually includes the option to install if you wish to.
Edit: A suggestion. If you decide to continue installing, please select the "something else" method. The install will bring up the partitions that presently exist on your hard drives. It appears that you have a SSHD and a slower HD. In any case, the something else method will show you partitions where your former install still may reside. Please make notes on these partitions, ext4, swap or whatever formats used - their size and other details. Please post them here.
Also see this guide - Install Ubuntu (Mate) using "Something else" method