Pre-ordered and thus gained BETA access earlier today. I've been awaiting the chance ever since I saw the pre-BETA video you guys released. 
I loved FE, but had some issues with it (as many people did) and I like the vast majority of what I'm seeing even in this BETA.
Alright, this is what I've noticed after playing for about two and a half hours:
1) Music does not play when you create a new faction. I didn't test out the other 'tools', so I'm not sure if they are also bugged like this or not.
2) Clicking on the upper left corner army icon doesn't always zoom to the army on the map. (this is just annoying when you play close-up like I do)
3) Sometimes the turn-'til-reached indicator doesn't disappear after you station an army in a city. This is only fixed when you move another army (or the same one, I suppose) outside of the city limits. [I really like this feature, by the way. Very helpful. Once bugs like this one are fixed, it will be an indispensable addition to FE/LH]
4) You REALLY need to allow the player to either discard or sell the "Warg Hunter's Note". It's REALLY annoying to have it just sitting in your inventory doing nothing but taking up space.
5) Icon for "Life Archmage" ability works, but the icon itself looks strange, as if it's broken.
Personal suggestion: I REALLY, REALLY, REEEEAAAALLLY want the Dynasties placed back into this series. I think it was one of the best RPG related bits to WoM and I miss it terribly. I have modded FE so that the buggy Dynasty system is enabled, but it would be absolutely amazing if you guys could actually implement it. [Also, I know that I'm not the only one who feels this way. Not by a long shot.] My personal wish for the Dynasty system is that you are able to marry off your sons/daughters to other champions in your service as well as to other sons/daughters of other sovereigns.
Edit/Additions as of later on 03/22:
1) I'm not certain if it's a lack of understanding on my part of the new battle components/tactics, but the Dodge rates in nearly ever single battle I have thus-far encountered are RIDICULOUS. As in, every third attack will actually hit, even when exploiting the supposed hit rate boost that new swarm mechanic gives you to the best of my ability. This creates very frustrating battles, which drag on for far longer than I'd like. Also, the ability Double Strike is just as unlikely to hit, even on the first strike of the series. I figured this issue out. I started looking closely at the enemy units I was fighting and found that their initiative was higher than my own; so, that problem is settled.
2) The spell "propaganda"'s effects are not counted in the turn-by-turn acquirement of gold. This is particularly bothersome to those of us who prefer playing a more magic-based kingdom/empire. This bug seemed to have been fixed in version 0.51. Thank you, Stardock! 
3) Sell prices are ALWAYS 6 gold. This is very irritating when you attempt to sell things you find in loot by mid and late game, because such items should be worth far more than that... This bug also appears to have been fixed in version 0.51.
4) The icons for level 2 and above upgrades to cities expand strangely when moused over...
5) A glitch occurs occasionally where another kingdom/empire will warn you of another's growing power... Every Single Turn... [ this NEEDS to be fixed...] and will stop and start without any apparent cause. This bug appears to have been fixed by the latest update (0.51)
6) When you rush production, sometimes the gold requirements include decimals. Is this on purpose? It just looks a little odd. [it's just a little quirk I noticed]
7) Kingdoms/Empires will demand tribute/payment (usually done so that they won't bother you for a few turns) even if they are at war with you and haven't offered peace/you haven't suggested it
8) The game crashes more often than not when you go to start a new game. Might want to look into that... Haven't had this happen since the update. (0.51)
Edit/Additions as of 03/26:
Biggest things...
1) Sometimes when you recruit a champion, their icon will appear in the upper right hand corner like it is supposed to, but you cannot actually move them, see them on the map or even truly select them. This is a big bug that must be fixed before the official release. VERY annoying. [Note: this usually happens when my sovereign is on a tile that has some kind of un-traversable tile one space away... even if there are open spaces around.
2) The achievement for defeating Vetrar does not trigger after you defeat him in game (Steam Achievements).
Less important, but still quite noticeable...
3) Occasionally, there won't be an actual animation (or attack numbers) from the attacker or defender, but a counterattack will be launched.
4) This bug actually happened to me a lot in FE, and is really just a visual/particle effect issue: a lot of times when you move a unit that has a hasted status indicator/yellow ring(s) swirling around it, the special effects/particle effects remain behind in a tile where the unit used to be. [Note: This doesn't always happen, but I have yet to figure out what exactly is triggering it...]
5) The icon for militia sometimes appears incorrectly, appearing to be just light and shadow without any character(s). [Note: This issue resolves itself after upgrading them to better armor and weaponry than they were originally given]
6) Sometimes Wisps visually disappear -- aside from their health bar -- during combat [Note: I haven't figured out what could be triggering this, but I'll continue to watch for it.]
7) Sometimes when fighting a battle with an enemy that is immune to a particular type of damage (I've only noticed it so far with immune to magic and immune to physical) the battle/action/attack forecaster that appears in the bottom left hand corner when you mouse over an enemy doesn't always show when the enemy is immune.
8) (Another bug that happened to me frequently in FE continues here) A lot of times when you heal a unit, the increase in health is not show in the health bar above the unit, only in the health bar in their full status when you click on them or mouse over them. [Note: This is exclusive to the basic Heal spell]
Edit/Additions as of 03/27:
1) (This happened to me in FE on occasion as well) When you tell a unit to move out of a city, occasionally they will move out to the desired spot, but then just head right back into the city as if you had ordered them to. This cycle is only broken when you choose another unit to move and it does not get stuck in this loop. [Note: I have no idea why this triggers, but I'll make sure to keep an eye on what's happening in the game and will add the apparent cause here if I can see one.]