ExFAT vs NTFS
exFAT  and NTFS  are filesystems created by Microsoft. These filesystems support storing files much larger than what Microsoft's old FAT32 filesystem supports. FAT32 has a file size limit of 4GB, while NTFS and exFAT are designed to support 16 EB (16.000.000 Terabyte).
NTFS has much more features than exFAT, "such as improved support for metadata and the use of advanced data structures to improve performance, reliability, and disk space utilization, plus additional extensions such as security access control lists (ACL) and file system journaling."
NTFS is faster than exFAT, especially on Hard Drives. exFAT is an incremental improvement over the old FAT32 filesystem, and has a simple file storage algorithm which performs well on Solid State Drives, but not Hard Drives.
Performance is largely dependent on what kind of data is being stored and how blocks are allocated in the filesystem. exFAT has better performance for lots of smaller files because of a smaller overhead for file allocation blocks.
 Cross-Platform Support
NTFS is fully supported on Windows versions newer than Windows 2000. exFAT is supported on Windows versions newer than Windows XP SP1.
Mac OS X has read-only support for NTFS, and read-write support for exFAT as of Lion. Apparently, the native NTFS driver on Snow Leopard had an unsupported way to switch on write support, but its stability is questionable. In Lion, Apple decided remove any hint of write support for NTFS.
You can buy commercial drivers for the Mac from Paragon or Tuxera. There is also an open source version of Tuxera called NTFS-3G that relies on FUSE, but the NTFS-3G driver is outdated and slower than the commercial versions.
Linux also has NTFS support in userland via FUSE and NTFS-3G. Unlike OSX, modern Linux systems usually make it practical to write to NTFS-formatted hard disks (or partitions) in addition to reading. A GPL implementation of exFAT has been released by Samsung, it was originally unintended, but they have since fully released it. It appears in various repositories as exfat-fuse and exfat-utils.