What difference do gems make in enchanting weapons?

Currently, I'm enchanting a Glass Bow (Exquisite) with Fire Damage. I have a Grand Soul Gem and a Petty Soul Gem available.

When selecting the Grand Soul Gem, the weapon has the following stats:

• Damage: 30
• Weight: 14
• Value: 1702
• Burns the target for 43 points. Targets on fire take extra damage.

When using the Petty Soul Gem, these are the stats:

• Damage: 30
• Weight: 14
• Value: 1372
• Burns the target for 43 points. Targets on fire take extra damage.

I've also noticed that the charge level of the weapon is just slightly less.

Since I plan on actually using this weapon, I really have a hard time seeing any worthwhile difference here. The only changes are:

• Value is lowered. (Not relevant to combat.)
• Weapon is slightly less charged. (Very small difference, and easily resolved by charging with another low-level gem.)

I could see using the higher gem perhaps if I wanted to go straight to market with this weapon. But other than that, is there really any point in using a higher-level gem to enchant weapons you actually plan on using?

The difference is how big the maximum charge will be. So, if you use the petty gem, you will have to recharge the weapon all the time. If you use the grand gem (assuming it contains grand soul), you will have to recharge it only rarely.

Source: Skyrim:Enchanting on UESPWiki:

The type of soul gem affects the amount of charges on a weapon or quality of the enchant on apparel.

The gem doesn't matter.

The soul inside the gem does.

You can get a lot more out of a grand soul than you can out of a petty soul -- many more charges (attacks) for weapons, much better stats from armor. A gem can also fit in any soul its size or smaller.

I'd suspect, from the information given in the question, that your grand gem has a lesser or common soul in it, so you're not going to see much difference between that and your petty soul gem with a petty soul in it.