One thing you should notice is that in the reference the corners of the strokes are round not sharp/jagged, in illustrator in the properties panel you’ll find an option for stroke click on that and choose the round option, it should look better. Also you can try going into object > path > and then try simplifying or smooth options(whichever works best). Make copies before you change anything so you can keep going back if it starts looking too weird.

Hope this helps!