Conquest lets you know right off the bat it's not pulling punches. Definite difficulty upswing from Birthright and I'm just through the first 3 chapters. Part of that is the reverse difficulty curve of Birthright, but it's a definite upswing even still. Was going to knock it down to Normal after playing Birthright hard, but I'm having a lot of fun so I'll stick with Hard for now.
The difference in difficulty (which I feel is a bit exaggerated though this may come from the lack of grinding when I first played Birthright) comes from three main factors:
1. Lack of Hoshidan weaponry and rods so healers are more in danger than they would be in Birthright and mages are even less useful than they were in Birthright. What can be considered the classic Fire Emblem weaponry is hard to go back to after the more helpful ones featured in Birthright, ease of being able to purchase Beast Killer, Armorslayer and other special weaponry aside.
2. Conquest is more consistently challenging in terms of a difficulty curve. Birthright's difficulty fluctuated from chapter to chapter: You could go from three relatively fair maps to one "oh wow this is tough, reset, reset, reset" map while Conquest goes from a challenging map to a slightly more challenging map.
3. Limited ability to grind gold and experience points. Invasions and paralogues do allow for additional experience to be earned, but it's not as forgiving nor as plentiful as Birthright.