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Are any of the books from Mongoose Traveller 1st edition mechanically compatible with 2nd edition?
The kind of compatibility I am thinking of means that I should be able to use rules or stats presented in a 1e book in a 2e game, ideally without changes. I would also call a book compatible if there were simple rules one could apply to make the system work.
If yes, which books are compatible?
Only CT, MT, and MGT are inter-compatible to each other, and even then, only characters really are.
Classic Traveller: baseline. 1-6 skills per term (1-2 Term «extra on 1st term», 0-1 Commission, 0-1 Promotion, 0-1 Service, 0-1 Rank, ), average about 1.6, stats 2d6. Attribute raises cost 1 skill level.
Generally, most characters have a level 2 skill or 2, and a lot of level 1's, tho a single skill at 3 or 4 isn't actually rare.
Mega Traveller: 1-8 skill levels per term (1-2 Term, 0-1 Position, 0-1 Promotion, 0-1 0-3 Special Duty «bonus possible on Position, promotion, special duty», 0-1 Service skill, 0-1 Rank skill), average about 2.5. Stats 2d6. Attribute raises cost 1 skill level.
Average skill level per skill is about the same as CT, due to more non-weapon skills; due to many more pick-from-list skill receipts, it's possible to get skills up to level 6 for many starting characters.
Traveller: The New Era: 1-8 skills per term (1-4 term, 1 Hobby, 0-1 Comission, 0-1 promotion, 0-1 Special Duty); Term skills per term starts at 4, and drops with age. Attributes 2d6-1, stat gains cost 2 terms' hobby choices.
Conversion from CT/MT in back of Survival Margin
No conversion needed for Twilight 2000 Second Edition nor Dark Conspiracy, but note that those always give 4 skills per term.
T4 aka Mark Miller's Traveller: 4-9 skills per term. (4 for term, 0-1 Commission, 0-1 Promotion, 0-1 special duty, 0-1 service skills, 0-1 rank skills). Attributes 2d6, attribute raises cost same as 1 skill level.
Note that skills are much cheaper than CT/MT; younger characters on par with TNE characters except for attributes.
Mongoose Traveller, aka MGT or MGT1: 1-5 skills per term (1 term skill, 0-1 from events, 0-1 promotion, 0-1 rank, 0-1 service skills), averaging about 2 skills per term, plus 1-3 skills per character at end of generation (0-2 from connections, 1 from party package) plus lots of level 0 skills (5x level 0 skills in first term, 0-5 background level 0 skills). Attributes 2d6. Stat raises 1 skill level each.
Most characters will have lots of level 1 skills; a smattering of 2's and 3's. Due to a different skill list, compatibility with CT and MT is less than 1:1, but MGT and MT characters in the same party are not a terribly big issue.
Mongoose Traveller Second Edition, aka MGT2: Same scaling but slightly different skill list from MGT1.
GURPS Traveller: Stats purchased in range of (typically) 7-18, but readable as if generated on 3d6 or 2d6+6. (IE: D&D kind of range, rather than CT's 2d6.) Skills bought with points. Totally incompatible characters.
T20 Traveller's Handbook: Characters are comparable with D&D 3.0. Totally incompatible characters.
Hero Traveller: Characters are standard Hero System characters, with Traveller specific setting packages.
T5: Characters are similar to T4, but the skill system works differently. Tasks are the same basic process as T4.
CT has two systems: Bk2 and Bk5
MT has one.
TNE has two fully compatible: FF&S1 and Brilliant Lances.
T4 has several: QSDS, SSDS, FF&S2; all are inter-compatible.
MGT has two: core and MGT Bk 2; the latter is for big ships (over 1000Td)
MGT2 has one, and it's not the same as MGT1.
T20 has one, and it's a CT Bk5 derivative.
GT has at least two, plus GURPS Vehicles, from which both were derived.
HT has one, it's a CT Bk 5 derivative.
T5 has one, it's a CT Bk 2 derivative.
CT Bk 2 is similar to MGT core for design, but combat is very different.
CT Bk 5, T20, and HT designs are all close enough to simply look-up and rerate for the new system.
MT's USP ratings are scaled identically to CT Bk 5, but the underlying design system is incompatible. This means one can easily use CT Bk5 and T20 designs in MT by ignoring the design rules, and just knowing how to read the ratings. And vice versa. MT's ship combat rules are literally just a task-system adaptation of CT Bk5's combat system.
TNE and T4 use the same ship and vehicle design rules; FF&S2 includes rating for T4, and extra options. FF&S 1 and BL include rating for TNE. SSDS and QSDS are built from FF&S1, producing legal but non-optimized designs, and rate for T4.
MGT core is similar to CT Bk2, and CT Bk 2 designs usually come across pretty well, but need to be rerated, and often have surplus space left if redesigned. MGT Bk2 is designed for larger ships than MGT Core allows, and produces MGT Core compatible designs, but they are incompatible with other editions.
MGT2 has ship design only in MGT2 High Guard. Process is similar to CT HG, but system percentages are closer to MGT1 in most cases. Ratings compatible with MGT1. It covers large ships.
T5 ACS is a system unique to itself. It resembles CT Bk 2 and MGT. No large ship rules are yet released.
CT: No ground vehicle design rules, per se... the minis game, Striker, is CT compatible, and includes design rules, so if one swaps out Bk1 combat for striker, striker vehicles can be used. (Azhanti High Lightning is a boardgame version of the striker combat mechanics)
MT: Striker designs can be used with mostly the same ratings, but hits need to be figured from the design. MT designs include everything needed to drop them into Striker, however...
MGT1: The Vehicle Design System was originally in two supplements, and a revised version was published in a combined supplement. Incompatible with all other editions, and the revision made major (but subtle) changes to the system.
MGT2: not yet in print. "Being worked on" was the last I heard from Matthew Sprange. Do not expect compatibility with other MGT vehicle design systems.
TNE: FF&S1 covers vehicle designs. FF&S 1 designs can be rerated for T4
T4: FF&S2 covers vehicle designs. FF&S 2 designs can be rerated for TNE T4 Emperor's Vehicles has an alternate vehicle design system rating only for T4. It doesn't convert to TNE well.
T5: T5 has its own vehicle maker, unrelated to anything else in the Traveller line.
CT, MT, TNE, T4, T20, and HT all use the same mainworld generation rules, with very slight variations, so all mapping and world data is compatible.
T5 expands the range of generatable worlds by conditional "exploding rolls" for certain elements, and adds additional resource data. Again, not all T5 generatable worlds are CT-generatable, and vice versa, but the data is useful across the editions.
Note that MGT and MGT2 reorder the sequencing, making Starport dependent upon population, and include several optional modifiers to rationalize certain physically improbable situations (EG: removing Standard atmospheres on worlds with ≤2400 km diameters). The result space is thus different. Some CT/MT/TNE/T4/T20-legal worlds are impossible to generate in MGT 1 or 2, and some MGT legal worlds cannot be generated with MGT. Furthermore, MGT redefined the meaning of Size 0: other editions, it means mainworld is asteroid belt, while size "S" is a world under 800 km diameter; in MGT, size 0 is a world under 800 km diameter; The data is largely usable across the editions.
Expanded system generation is in 4 groups:
Combat stats and mechanics are so different that simple crossover isn't readily practical; one can, however, pretty much pick one's choice of CT Bk1, Striker/AHL, MT, MGT, MGT2 or T4 combat and use it with the others. There are some hiccups: CT has no task system, and thus, due to skill level differences, T4 isn't really compatible with CT Bk1 combat, and you need to borrow a task system to use any non-CT combat system with CT. T5 scales similarly to T4 - gear is easier to port than the combat system.
Setting Information - except for CT, the setting is built into the core books to some degree. CT, it's in the expansions (Books 4-8; Supplements 3, 4, 5, 8, 10, 11, and to a lesser extent, 12 and 13; Adventures 2-3 & 6-12). MGT, a minimal sketch is in the core, and most of it is in a separate "Third Imperium" trade dress supplement line. MGT2 is likewise only a minimal sketch in the core. T5 present only a brief schematic in the core.
Older products were released under a Traveller Logo License. It was sunset in 2016; many have been rereleased electronically without the license. It indicates MGT1E compatibility.
The "TAS license" is available for MGT2, and indicates compatibility with MGT2 only.
Cephus is built on the MGT1 SRD. It is, fundamentally, MGT1 retooled to generate more CT-Like characters. Combat, Ship Construction, and World Generation all parallel MGT1; there is no expanded system gen nor ground vehicle design. Many MGT1 materials are being retooled slightly to work with Cephus. The SRD, while bare bones, is fully complete.
You can certainly use 3.0e and 3.5e books together. There were many changes between 3.0e and 3.5e, mainly focusing around balance issues. Unfortunately no intentional balancing was done in 3.0, and as such one CR 11 monster would be easy, and another CR 11 monster might be a lethal encounter.
The biggest individual changes are.
The official change guides can be found here.
Best Answer
Simple answer, none and all of them. None remain 100% compatible; all remain broadly usable.
There are a lot of books released for Mongoose Traveller 1st edition (MT1e), so I couldn't put together a list as requested solely because it would have to be pain-stakingly researched to give you an accurate answer.
Right from the start of the open 'playtest' last year, Matt Sprange noted on the Mongoose Forums that while the core principles of the old system would remain, the design team would not let any of the old material hold them back from making changes. If a change would make the game better, they'd make it - backward compatability be damned.
On the other hand, they were not setting out to intentionally invaidate any of the old material - and if you own one or more books from MT1e - or, indeed, other versions - you will not find them rendered worthless. You will need to do some legwork understanding the changes and deciding what to keep and what to go with.
Broadly speaking, the stats for people and places have remained the same. The crunch for ships and ship weapons got a bit of a tweak as did the system for large scale battles. World building has remained the same as MT1e; character creation, especially skills, has undergone some slight adjustment.
As I say, you will find the old books generally usable - but Mongoose did not hold back from making changes in MT2e to 'make it better'.