Should I choose TI, Microchip, Arm?
ARM doesn't make any actual chips. They just license their design to chipmakers. Microchip (Atmel) and TI (among others) both make chips based on ARM architecture.
What is the difference except voltage operating level?
Peripherals: Different chipmakers will include different combinations of peripherals on their microcontrollers. One or the other might be a better match for your application.
Tools: Each chipmaker provides their own development tools for their chips. Third-party tools are also available for many microcontrollers. You might find you like one set of tools better than the others, which would be a point in favor of using the chips supported by that tool. FWIW, TI's free tool is based on Eclipse while Atmel's (a Microchip subsidiary) is based on Visual Studio. Microchip's tool for their legacy parts (pre Atmel acquisition) is (AFAIK, I haven't used it) proprietary.
Performance: Different chipmakers might offer microcontrollers with better or worse performance by some metric, such as clock frequency, power consumption, etc.
Architecture: As mentioned before, many chipmakers produce ARM-based chips, and these are very popular. But proprietary architectures are also still out there. You might find you prefer one or another, or that one is particularly suited to your application. Microchip's PIC architecture, for example, is a Harvard architecture, which has some advantages for keeping code and data separate (preventing code from being corrupted by rogue instructions meant to act on data).
Support: Some vendors might make it easier to contact an applications engineer to get help when you have trouble with their parts. Others might only provide a bulletin board system where users can help each other. Others might provide even less.
Price: Different vendors have different strategies for pricing and marketing. One might offer very low prices but only for very high volume customers. Others might have higher prices but give good support to smaller customers. Etc.