The difference between UEFI Boot and Legacy boot is the process that the firmware uses to find the boot target.

Legacy Boot is the boot process used by BIOS firmware. The firmware maintains a list of installed storage devices that may be bootable (Floppy Disk Drives, Hard Disk Drives, Optical Disk Drives, Tape Drives, etc...) and enumerates them in a configurable order of priority. Once the POST procedure has completed, the firmware loads the first sector of each of the storage targets into memory and scans it for a valid Master Boot Record (MBR). If a valid MBR is found, the firmware passes execution to the boot loader code found in the MBR which allows the user to select a partition to boot from. If one is not found, it proceeds to the next device in the boot order. If no MBR is found at all, the user is presented with the famous "Please insert system disk yadda yadda yadda"

UEFI boot is the boot process used by UEFI firmware. The firmware maintains a list of valid boot volumes called EFI Service Partitions. During the POST procedure the UEFI firmware scans all of the bootable storage devices that are connected to the system for a valid GUID Partition Table (GPT). Unlike a MBR, a GPT does not contain a boot loader. The firmware itself scans the GPTs to find an EFI Service Partition to boot from. If no EFI bootable partition is found, the firmware can fall back on the Legacy Boot method. If both UEFI boot and Legacy boot fail, the famous "Please insert system disk yadda yadda yadda" will be presented.

In general, UEFI boot is more desirable. 

UEFI is a little bit faster to boot up, but will perform the same once the OS is up and running.