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Working out D&D 5th edition rules

We are newbies trying to get the rules down, and have some questions we haven't been able to resolve yet. We may have more, but so far we have mostly been able to work out our own answers. Here are a couple of places we haven't been able to yet.

1) Wizard page 114 has a Spell Attack modifier , but the spells I have looked at all specify attack damage without using this modifier or mentioning it. I can't determine what it has to do with anything.

2) I'm not sure I understand the purpose of the different types of damage, i.e. if I take 3 cold damage as opposed to 3 fire damage, I lose 3 hit points, but what else? Is the only purpose that maybe I have defense against fire damage, but not cold damage?