Why did Snape claim that Tonks's Patronus looked weak in the 6th book? Did he know how she felt about Lupin? If so, how? Did he read her mind? And why did he say that given what his Patronus was?
He clearly recognized Tonks' patronus as Lupin, whom he disliked intensely. It's unclear whether he was merely sneering at the idea of Lupin being anyone's Patronus or was mocking Tonks herself for falling in love with a werewolf.
As for his own Patronus referencing Lily, I'm sure he'd be insulted at your insinuation that this could be even remotely comparable. (Lily == perfect, Lupin == weak. Makes perfect sense from Snape's view.)
Interestingly, when Harry witnesses the scene, he thinks Tonks' Patronus is Padfoot, but Snape's derision is just as believable.
He said it because her Patronus matched someone he hated.
Snape wasn't disapproving of the fact that her Patronus changed because she was in love. He was disapproving of specifically who she fell in love with.
“He shut the gates in her face with a loud clang and tapped the chains with his wand again, so that they slithered, clinking, back into place. ‘I think you were better off with the old one,’ said Snape, the malice in his voice unmistakeable. ‘The new one looks weak.”
- Harry Potter and the Half-Blood Prince, Chapter 8 (Snape Victorious)
He wouldn't have needed Legilimency to figure out what was going on. He would probably know the same thing Lupin explains to Harry, that a Patronus can change due to emotional upheaval, even if he didn't personally have a corporeal Patronus before his was a doe like Lily's.
“Tonks’s Patronus has changed its form,’ he told him. ‘Snape said so, anyway. I didn’t know that could happen. Why would your Patronus change?’
Lupin took his time chewing his turkey and swallowing before saying slowly, ‘Sometimes … a great shock … an emotional upheaval …”
- Harry Potter and the Half-Blood Prince, Chapter 16 (A Very Frosty Christmas)
After seeing that her new Patronus was a wolf, it's not a far logical leap to connect it with Lupin, the werewolf. In addition, he might have actually seen Lupin's Patronus and recognized Tonks's matched it. It's also possible that Snape might have heard about her feelings for Lupin somehow - the Weasley parents certainly seemed to know.
Harry suspected Tonks was in love, and not only was he not a Legilimens, he wasn't even particularly observant. Even if Snape managed to get it entirely wrong, like Harry did, he could have thought it was Sirius, who he also hated.
“You don’t think she can have been … you know … in love with Sirius?’ Hermione stared at him.
‘What on earth makes you say that?’
‘I dunno,’ said Harry, shrugging, ‘but she was nearly crying when I mentioned his name … and her Patronus is a big four-legged thing now … I wondered whether it hadn’t become … you know … him.”
- Harry Potter and the Half-Blood Prince, Chapter 21 (The Unknowable Room)