The manual states (page 16):

"The crew has four experience levels:
• Rookie
• Regular
• Veteran
• Elite"

Yet at the end of the campaign, the maximum that my ships have reached is Veteran. How can I reach Elite? Do you get more experience at higher difficulties?

from memory its the other way around i.e, lower difficulties = higher experience

It seems to me that difficulty does indeed influence experience. In the files at

Nexus The Jupiter Incident\universe\missions\EPISODES

in the main mission files (for example 052___TheChase2.mission), we have code like this:

If(MissionObjective(2)=#OBJECTIVE_COMPLETED,
ChooseFirst(
M.diffLevel=1,
uGetNPC(#NPC_Player):XP := uGetNPC(#NPC_Player):XP + 1;
,
M.diffLevel=2,
uGetNPC(#NPC_Player):XP := uGetNPC(#NPC_Player):XP + 2;
,
M.diffLevel=3,
uGetNPC(#NPC_Player):XP := uGetNPC(#NPC_Player):XP + 3;