These are two editions I like and know well. And I'm at a loss with even where to start trying to explain the differences to someone thinking they're similar.
The trouble is that every edition after AD&D 2nd edition wasn't a revision, it was an entirely new game that just used similar labels and a few somewhat related mechanical pieces. Between Basic/Expert D&D and D&D 5e are six major editions, three of which were basically throwing out everything from the previous edition and writing “D&D” from scratch but with similar names. Where BX is recognisably the same game as Gygax and Arneson originally published in 1974, just refined in presentation and clarity of materials, 5e is effectively the great-great-great-grandchild of Original D&D, with all the differences (and generational “differences of opinion”) that implies.
To really understand the differences between BX and 5e, you have to understand the evolution of the various game lines over about three decades. So instead of trying to write a book here (which probably would need to exceed the post length limit), I'm going to give you a road map to understanding the chain of design change and repurposing that has gone on while you've been away.
I know, I know, you just want “a list of equivalents from one to the other.” That's something that could in theory be written, but it would be more misleading than helpful. Whole paradigms of design and even what the point of playing a game of D&D is have come and gone, leaving their mark on these terms that seem familiar. There's a concept when learning a second language that is similar: “false friends”. False friends are words that look related, but have no meaning in common and are effectively seductive conceptual traps for the learner that do more harm to learning than good. The history between BX and 5e have created a game riddled with “false friends.”
Fortunately, we have accumulated a series of questions about the changes between D&D editions. Reading them and their answers can be a map of the way the games are related (and how they're not), giving you an idea of just how different the moving pieces of 5e are from what they look like.
Comparing D&D mechanics between 2e and 5e
This is probably a good place to start. It won't explain a lot of the change in the middle, but it will give you a “bird's eye view” of how much has changed between AD&D and D&D 5e. This should establish a bit of a sense of how little is still the same — knowing how much there is to know (or, how little one actually knows already) is the first step in understanding any subject.
The answers to this question also hint at just how much cultural change within the player and designer communities has happened since BX was in print.
Unfortunately, there is a gap in our roadmap: there is no Q&A on the differences between AD&D and 3rd edition. We'll have to jump straight into the changes between 3rd and 4th:
What are the differences between D&D 3.5 and D&D 4th edition?
This one is mostly going to sound like Moon Language to you at first. Fourth edition is a very different game than BX, and 3e/3.5e are superficially similar but exceedingly different under the hood and in play than BX. This Q&A will mostly serve to give an appreciation for how little it is safe to assume about similarities between editions you know and any other edition. Just the three sentences about multiclassing in the first answer there is enough, if I put myself in the mindset of only being familiar with BX, to sort of blow my mind about how alien 3.x and 4e are compared to the original game. This isn't Kansas anymore.
There are a couple other questions that provide the jumps between 3.x and 5th (skipping over 4th, which was a contentious edition some people ignored), and the difference between 4th and 5th:
In general, what I hope reading these has provided is some insight into what the current terminology of 5th edition is building on, changing, and redefinding, and how “D&D” has a lot of baggage that it didn't have back when BX was new.
Now the best advice I can give is to approach D&D 5th edition as if you know nothing about D&D, and read it like a fresh, unknown game. Assume nothing about similarities; assume everything is going to be different. In the end, you can still get some of the same kinds of adventures out of it, but how it gets there is different in ways that are not all obvious.
All that said, here's a short cheat-sheet for the points you mentioned specifically in your question:
Races and classes are similar in overall purpose, though specific classes have had many changes in what their core purpose are, and there are new ones that will be unfamiliar. In particular though, notice that the concept of allowed class/race combinations (from AD&D) has disappeared: any race can be any class. The BX idea of race-as-class is nowhere to be seen.
Hit points are vaguely the same. However, “hit dice” are going to be confusing: they're now both what you roll to determine your hit points increase when you gain a level, and also (entirely separately) something you “spend” to heal when you rest. Hit point totals have inflated since BX: each point is worth less, and there are more points floating around.
Ability scores serve the same purpose. Bonuses from ability scores are standardised though, and bonuses are bigger and gained for lower scores.
Saving throws have gone through several personality changes over the editions. (In 4e the name was used for something completely unrelated!) The 5e concept is inherited from 3.xe, which replaced the five types of saves from BX with three saves based on three ability scores, one each for situations where mental resilience, physical resilience, and the ability to quickly get out of the way are challenged. 5e standardised this by just making each ability score its own source of a way to roll a saving throw.
THAC0 has been out since AD&D — when 3e was released in 2000, the math was kept the same but rearranged so that subtraction didn't happen and you always counted up. That meant that AC had to be flipped around, so now it starts at 10 (as before) and goes up, and you get more competent by having bigger bonuses instead of a lower THAC0. This lead to “bonus inflation” where PCs would have to-hit bonuses in the +20s and +30s, making that longsword +1 seem pretty worthless. 5e has reined in some of that, but you'll still see way more bonuses than you're used to.
BAB (Base Attack Bonus) is from this “everything is a plus” design, and conceptually it's basically 20 − THAC0, though the numbers have changed a bit so it doesn't line up exactly. However, 5e has done away with BAB (it was a 3e/4e concept) and has replaced it with a more standardised Proficiency Bonus, which applies to both combat and skills in which a PC is proficient. To combat bonus inflation, Proficiency stays in the low single digits.
- As an aside, magic items have gone through some major life changes. (At one point they were actually in the PHB and were freely available for purchase and crafting by PCs, so much so that not buying magic items made your character too weak to adventure with others of your level.) 5e has attempted to bring magic items back more in line with how they were in AD&D, so they're rare again and not required anymore. They're still more prevalent than you might be used to, but a longsword +1 is worthwhile again.