Great. The problem is: Did you remember to check your character sheet or are you just working off of memory? If you bonus to hit is +8 and you use that +8 on 80% of your rolls are you going to even remember to apply the +2 because this round you are tripping instead of doing a normal attack? Where is the note on your character sheet? Right beside your weapon or on a page of notes on the back? If you put it right beside is it big enough that you'll see it when you check your character sheet? Even if it is, will you remember that when making a trip attempt you need to look at your modifier AND the note below it or will you just check the modifier absentmindedly and gloss right over the note?
Also if there are 4 more notes below that one saying +2 against Orcs, +2 against people wearing red, +2 against Mimes, +2 vs Sandwiches....and you almost never look at any of them because you rarely fight sandwiches, are you going to forget to check that note the day you actually fight sandwiches?
Even if you do remember to apply it, now there's a modifier you MIGHT have forgotten so now the group has to ask if you've forgotten it, just in case, since they want to make sure you are applying every possible benefit. If you get angry at them for reminding you all the time, they'll tell you they were just trying to be helpful and there's no need to yell. Then forget that you yelled at them and start reminding you again next turn.
All this fails to ask the fairly legitimate question: Is it worth all the hassle just to have a +2 bonus to trips?
You took what I was trying to say to an extreme, so it seems fair I do the same.
Then why have the other weapons in the game?
That's really what it boils down to for me. If there's no difference in what they do, why not just say "little weapons do 1d4, average weapons do 1d6, large weapons do 1d8, and big honkin' two handed weapons do 1d12"? Then just have the player fluff what the weapon is because it doesn't actually matter in play.
I find that pretty boring, and I it also annoys me because, as I said earlier, I dislike when choices really aren't choices... whether that's because they don't matter or whether that's because one is obviously so much better that it makes no sense to choose the other ones. When it comes to D&D weapons and equipment, somehow the game occasionally manages to have both be simultaneously true. I personally dislike that. I understand that others may not have a problem with it, but I do.
As for your question about whether I'd remember things... personally, I would. There's plenty of empty space on my character sheet. If I really have to, I'll use a scrap piece of paper or a notecard. For what it's worth, I was (and I'm not at all making this up) where Pelor had been brainwashed into believing he was the "god of 'sammich,'" so I likely would have been pretty happy with a weapon which got bonuses against sandwiches in that particular campaign. Now, do I believe that's something which comes up often? No, I certainly don't, but it amused me that you picked such a way to make your point since I had actually been in a campaign where it would have mattered.
More to the point, I'm not suggesting to add a slew of modifiers. I'd be happy with just making the multitude of weapons simply not seem pointless. If a +1 or -1 seems too hard to remember, simplify it further by using what is already a common 5th Edition element by saying "this weapon gives advantage on trip attacks." Personally, I paused from saying that as my original idea because I feel that's far too good, and I would believe it to be unbalancing (pun intended,) without some sort of cost. As such, I felt that starting with a minor bonus (such as a simple +1) as a base property for a few select weapons was the better idea.