So while the new site looks better than the old one, I'm not sure how much of an improvement it really is. Visually, yes. It's prettier. It doesn't bother me as much since I rarely book directly (most travel is for work through corporate travel agency).

• Remember Me option, as others have indicated, seems to work at random. Sometimes it remembers my name and MP# and just requires a PIN entry. Other times, it forgets everything.
• Under the check-in box, when I click the arrow next to "Upcoming Trips" when I have no trips eligible to check-in, it expands and says so. When I do (<24hr), the persistent behavior seems to be that the click does nothing. Then I have to remember the PNR, or just type my MP# in the box, and get redirected to another page to enter my MP# and PIN again before I get check-in.
• Geo-IP detection for logged in users is a joke for business travelers. If I fly to and from LGA 90%+ of the time, why are you doing geo-IP, and not my major market based on flight history/home address?
• Putting individual prices per each way when the user does a roundtrip is a stupid and useless way to display the information when the user is trying to price a trip. It makes me want to use third party tools like Hipmunk or Kayak or google Flights to find a flight. Then I book directly, but United still has to pay a fee to the third party for referring me there. (In the "tour the new united.com" link at the bottom of the homepage, they actually tout "directional pricing" as a feature )
• If I am logged in with a MP# that's gold, only Y or B fares should have the upgrade eligibility notation. M should not display "Eligible for Upgrade" for someone that isn't high enough status to get an instant upgrade. If the worry is that a fare would be upgradeable for another person on the same reservation that is 1K, then the text should say "Eligible for Upgrade (1K only)". I should not have to hover over the eligible for upgrade text.
• Information about the baggage charges should be transparent. A small link about the additional baggage charges is OK. It's just lazy that it takes you to a generic page about checked baggage, on which you must then select "Any flights" and then enter the city pair to find out what the fees are. Not as big of a deal for an elite, but just poor form. You know the city pair, you know the cabin, you know the date. At a minimum it should link to the page to check those fees with that info pre-selected - with the "status" modifiable so people with *G or MP Elite status can put that in.
• The fare rules link after selecting the flight (before continuing to select passenger info/payment) is ridiculous. it asks for a CAPTCHA to be filled, even if one is logged in. You enter the captcha, only to find a generic page for the fare code for the given city pair. This page has several sections, including large sections (e.g. surcharges) that list all potential things that could apply at any point. These take up multiple pages and within a section (e.g. surcharges) They list dozens of dates. Even more puzzling is that the URL for the fare rules is not something that has the fare code, and city pair, but the cart ID - so they know exactly what dates your trip is from, yet they spew out paragraphs of worthless garbage that is difficult to skim.
• For flight searches, the "recent searches" functionality seems to always display "no recent searches" even if I have searched a particular flight #/date or date & city pair.

Beyond that, I know that changing everything at once is hard and rolling out portions over time is easier, but the visual differences on commonly accessed functionalities (e.g. viewing a reservation, searching a flight by departure and arrival city, etc.) is pretty jarring compared to having a similar navigation across all of those functionalities.

tl;dr It looks nice but there's still a lot of issues I take with the new site.