What are the differences between MBR vs GPT vs any other partition scheme?

Differences between MS-DOS style Master Boot Record (MBR), the Apple Partition Map (APM), and the UEFI-style GUID Partition Table (GPT) are such:

Disk size

MBR and APM limit the usable disk size to 2 TiB (a partition can neither start nor end beyond the 2 TiB limit). With GPT, the disk can be up to 8 ZiB.

Partition count

MBR is limited to four partitions.

To get around the limit, one of the partitions is usually created as an "extended partition" which nestedly contains a series of "logical partitions". The most common scheme for this is an Extended Boot Record, though BSD systems often nest a BSD disklabel instead.

APM can grow up to 62 partitions; GPT can have at least 128.

Partition metadata

MBR partitions have a 1-byte "type" code, which is too small to be useful (most operating systems use very generic type codes and just guess the rest). Windows NT also introduced a 4-byte "disk ID" for distinguishing between multiple disks of the same model.

APM uses textual "type identifiers" (32 ASCII bytes), e.g. Apple_UFS. It also reserves 32 bytes for a descriptive partition name.

GPT, as its name tells, uses a 16-byte GUID for partition type, another GUID for identifying a specific partition (the partuuid), and yet another GUID for identifying the whole disk (similar in purpose to the MBR "disk ID"). It also reserves 72 bytes (UTF-16) for a partition label.

System architectures

Intel-based Macs can only boot from GPT drives, while PowerPC-based Macs only support APM.

"Regular" IBM-compatible PCs can boot any disk as long as the 0th sector contains a valid BIOS bootloader and the AA55h boot signature. This usually means MBR, but all GPT-partitioned disks have a "protective MBR" that satisfies this requirement.

PCs using UEFI are required support both GPT and MBR – again, as long as an UEFI-compatible bootloader is present. (UEFI keeps bootloaders in a FAT32 partition with specific type code.)

(Exceptions: Some specific BIOS versions reject the GPT "protective MBR" as invalid, due to bugs or misfeatures. Also, Windows will refuse to boot from a GPT disk on BIOS systems, or from a MBR disk on UEFI systems, due to reasons unknown.)

Boot loaders

In BIOS systems, the initial boot loader is part of the MBR. There can only be one bootloader in the MBR, resulting in conflicts when installing dual-boot systems. Due to the x86 DOS origins of the MBR bootloader, code is specific to the x86 architecture.

On the other hand, UEFI uses a dedicated FAT32 partition for bootloaders (potentially multiple) and other EFI tools. The partition contents can be easily managed from any OS.

Technical bits and pieces

• There is only one MBR or APM per disk, both starting at sector 0. GPT keeps a backup copy near the end of the disk.

A more detailed info related to MBR and GPT: Windows and GPT FAQ

I only know of three partition schemes: MBR, GPT, and APM. (These are the ones Apple's Disk Utility can create.)

MBR (Master Boot Record) is used historically by Windows (and, therefore, by most computer manufacturers). I'm not certain if you can boot a Mac from an MBR disk; I don't think so.

GPT (GUID Partition Table) is generally used in conjunction with Intel's EFI BIOS replacement. As such, it is mostly used by Apple currently in their Intel Macs. Any Mac running 10.4 or above can read a GPT disk; only Intel Macs can boot from them. If a computer has EFI, you can boot Windows on a GPT disk.

APM (Apple Partition Map) is what Apple used before the Intel switch. It's been used since 1989 in the Macintosh SE. Intel Macs can boot from and read APM disks.

I'd say the assertion you made in the question ("GPT is more useful than MBR") is incorrect: you can certainly boot more computers with an MBR disk than one using GPT. GPT is the way of the future, but it may take awhile for the major PC manufacturers to switch to EFI from BIOS.

Your Time Machine disk is almost certainly uses GPT, but that's totally irrelevant as you'll never boot from it.