I grew up in area with bears, and while I think this is a good creature I see a common mistake made about bears.

Black bears climb trees, brown ones don't.

Larger brown bears weigh enough that climbing a tree isnt a very good strategy, they may push them over to get at food but not climb.  These bears, I think, should have a charge special that knocks targets prone after moving a set distance, or a dash bonus action.  If you review attack reports, or are familiar with bear behaviore you would know they are very fast, capable of sprints almost 40 mp/h (60 kmh/h) or 50 ft a second... and thats up hill, down hill across slopes or uneven terrain;  its really quite a scary sight.

 Another option I might do is change the bite to a second claw, and if both claws hit the same target they are knocked prone and take bludgeoning damage from the bear landing on them, or then a bite in stead of the bludgeoning damage.

I totally agree but its D&D, and Wizard's haven't been very rules intensive with this edition so they probably won't care a ton about specificity and realism as long as its balanced.   

Brown bears are perfectly adept climbers, so the stat makes perfect sense.

https://youtu.be/hURgA_BNSGc

Based on this video, what kind of bear is best?

Can anyone explain the +6 to hit where the MM brown bear has +5?

Str of 19, for a +4 modifier.  +2 for Proficiency bonus = +6.

Yes, but in the MM it is the same stats aside from +5 to hit.  Which is proper? Is it for sure a +2 proficiency of is it a +1

Though in the video description it says they're both blackbears... sounds strange though. Cool Video though!

Brown and black bears were errata'd recently to have better to-hit on their bite and claw attacks.

MM Errata November 2018 (pdf)

That is quite simply not true. Brown bears do climb in trees,I've seen it happen several times over here in forests of Finland.

Bears.

Beats.

Battlestar Galactica.

Quote from Whambam >>

Based on this video, what kind of bear is best?

 That's debatable, there are basically two schools of thought.

How I miss the Dire Bear.....it is sad it was left out of 5e

I think they should probably have a better Intelligence. Bears are pretty good at figuring things out. At least 6.

This picture looks more like a cougar

I'm pretty sure all beasts have less than 4 intelligence 

Quote from Himmelfahrt_Bouldergut >>

This picture looks more like a cougar

 It's just the default for a beast type monster.

Is this the bear that you can use at level 2 druid? After we finish our first campaign, DM is gonna let us multiclass and make super op characters for our next campaign together, i'm making a Barbearian and need to know which bear is the one I can shapeshift into.

Quote from Horndrill >>

I'm pretty sure all beasts have less than 4 intelligence 

Not so. For example, the ape (regular or giant), baboon, clawfoot, crag cat, cranium rat, deinonychus, dolphin, fastieth, flying monkey, giant eagle, giant elk, giant octopus, giant owl,  giant rocktopus, giant sea eel, giant vulture, giant weasel, sea lion, tressym, or velociraptor. To be honest it was kind of a bummer when I realized their Intelligence made it impossible to awaken any of these.

Moon druids can turn into any non-flying/swimming beasts of CR1 or lower as of level 2. Just check the entry to see what CR it is and if it’s a beast, and you’re good to go!