My fighter exchanged a crossbow for a longbow at a Weapons Workshop he came across. The character has proficiency with Simple and Martial Weapons, with the Dueling fighting style, and is Dexterity modifier is +4.
By replacing the crossbow, would the attack bonus stay the same?
Also, the crossbow had a damage roll of 1d6 + 6 piercing damage.
Would the Longbow's damage be just 1d8 or 1d8 + 6?
Specifically, the Dueling fighting style requires you to be:
wielding a melee weapon in one hand
Longbows and crossbows are ranged weapons, and in addition to that, longbows require you to use two hands (see the two-handed property). Thus the +2 from the Dueling fighting style shouldn't be added to the damage of either attack.
The resultant damage for the longbow is then 1d8 + your Dexterity modifier, so for you it would deal 1d8+4 piercing damage.
The bonus on the attack roll is the same. For ranged weapons it is simply the Dexterity bonus + proficiency bonus (if proficient, which they are), plus any other bonuses they have from features (such as the Archery fighting style if they have it) or magic items.
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.
As listed on page 14 of the Player's Handbook,
For attacks with ranged weapons, use your Dexterity modifier for attack and damage rolls. A weapon that has the thrown property, such as a handaxe, can use your Strength modifier instead.
There's no reason your bonuses would change between a hand crossbow and a longbow unless one of them is magical or you have a specific ability that applies to only one of crossbows or longbows. There's no general rule that would cause them to have different attack or damage bonuses; the only difference would be the damage die size.
