Booking on United.com (016 ticket)
Pro
-Category bonus with United Chase Credit cards (ie- the Explorer Card earns two miles per dollar spent on United.com but not on Lufthansa.com, for example)
- Deal directly with UA on UA schedule changes
- UA change rules are generally more generous
Con
- Many partner flights will be booked as UA codeshare and this can make some things more difficult or impossible -- such as *A upgrades, seat assignments, actual partner fare class is hidden, ..
- While not a major thing, in irrops, codeshares can be an extra issue for the agent
= Due to plating restrictions, partners may not allow UA access to the partners best fares

Partner site
Pro
- Native flight numbers (No codeshare issues)
- Deal directly with partner for partner schedule changes
- Access to partner's lowest fares

Unclear
- partner flights will earn PQPs based on miles and multiple and UA operated flights will earn PQPs in the fare method -- which may be hard to determine in advance
Not hard rule which is better but a good chance for higher cost tickets PQPs on 016 ticket are better, and also for lower cost fares, you need to run the numbers.

Does not matter
-PQP earning & UA Lifetime mileage earning is unaffected by booking method
- On UA operated flight your UA elite benefits fully apply and you can use PlusPoints or miles to upgrade