Going by memory, I think the biggest differences are what rules didn't exist in 1e originally, but got added over time and all were included in 2e. I agree with you I think 2e is closer to BECMI including the Expert and Companion rules. I think Basic specifically is closer to 1e. For example, paladins and bards. Originally didn't exist in 1e but got added along the way and were included in 2e. Similarly Paladins weren't in the Basic rules, but got added with Expert and Companion, bards never existed in BECMI. Half-elves and half-orcs also, not in BECMI, not originally in 1e, but are in 2e and orcs got added late in the BECMI run with the Gazeteer for humanoid races.
It's been a long time, and I never played much 1e or 2e so I might be misremembering. Despite the name, I always found them overly simplistic compared to BECMI which I preferred.