Other answers have discussed changes (and/or lack thereof) to your attack roll, damage dice, and damage bonus, as well as possible misunderstandings in the sources of these bonuses (adding proficiency to damage, adding dueling bonuses, etc). At risk of going beyond the scope of your question, I want to point out two other changes - one subtle and one dramatic.
The subtle change: Range
Longbows have substantially better range than crossbows. Hand crossbows have a range of 30'/120'; light crossbows, 80'/320'; heavy crossbows, 100'/400'. Longbows, by comparison, have a tremendous range 150'/600' - 50% more than even the huge 18 pound heavy crossbow!
I mention that this change is subtle, because range is one of those things that doesn't come up in every game. Dungeon combat is often very close range, close enough that a light crossbow's 80' range is usually plenty. Adding to this, your table might not treat distances at all, in which case your range to target is completely immaterial. On the other hand, if you ever need to take out a scout as it flees on a horse, you'll be glad to have the extra range afforded by the longbow.
The dramatic change: Loading
All crossbows have the Loading trait. In short, this makes it impossible to use your Fighter's Extra Attack feature (unless you take the Crossbow Expert feat, which allows you to "ignore the loading quality of crossbows with which you are proficient").
Longbows, notably, do not have Loading. While an arrow from a longbow may deal the same as a bolt from a light crossbow, a skilled Fighter can let loose many more arrows in the same amount of time. This should not be overlooked and it will make you far deadlier as you advance.