As some one who has played in and/or run every edition except one (Mechwarrior: Destiny) here is my rundown on the editions.
First, you have to know that, all the pervious games suffered under the issues of "throwing out the baby with the bathwater", so there is little that stayed the same between editions.
Mechwarrior 1st edition
Pros: Easy integration with the TTG, Fast Character creation.
Cons: Little to no character variation, no Clan info. (pre invasion)
Mechwarrior 2nd Edition
Pros: Best integration with the TTG, Fast Character creation, intuitive combat system, very flexible design.
Cons: Has an all powerful attribute (Intuition) that powers just about every skill in the game, can be powergamed easy.
My favorite edition, and I am currently working on Mechwarrior 2nd edition advanced rules to bring some of the best parts from 3rd and 4th in to the game and it fixes the issues with Intuition,
Mechwarrior 3rd Edition (Classic Battletech RPG)
Pros: Introduced the Lifepath system allowing players to really get a feel for their characters and what they had gone through to get to the starting point of the campaign.
Cons: the random tables in the Lifepath system can brake a character before you even start the game, changed dice systems (2d6 to 2d10) making integration with the TTG a chore and a half, split the gunnery skill into 3 different skills really hurting Mechwarrior characters.
A Time of War (Mechwarrior 4th Edition)
Note: as with 3rd above, the name was changed due to a marketing issues where the computer games have the same name.
Pros: cleaner rules then 3rd almost to 2nd levels, returned to 2d6 system, expanded on the types of characters you can be by a lot.
Cons: removed the lifepath random tables instead of fixing them, made the life paths unbelievably generic, requires a spreadsheet to make characters using the lifepath system, point totals for character creation way to high to be practical (100 points just to get a Dex of 1 where 10 points would have worked).
Not a fan, but if you decided to go this route Daryk is your man.
Mechwarrior: Destiny (Mechwarrior 5th Edition?)
Pros: Has an interesting take on fast mech combat. (may steal some of it for MW2 Advanced)
Cons: To many to count.
Not going to say much more on this as it is frowned upon on the boards.
Needless to say if you are interested their is a board here on the game.