Would anyone like to have a go at explaining how and why the grammar of these two sentences differs?

A. I know students have problems with syntax.

B. I have known students have problems with syntax

At first glance, they look very similar, but, while 'A' clearly ends with a noun clause...

A1. I know that students have problems with syntax.

A2. I know that they have problems with syntax.

...what happens with 'B'?

B1. I have known them have problems with syntax.

B2. I have known them to have problems with syntax.

Why does the present perfect tense make such a difference? Does the meaning of 'know' change somehow?

Thinking back (a long time!) to when I "did" Latin, both constructions exist in that language. In sentence A, "have" is in fact (arguably) a subjunctive to indicate a subsidiary clause, while in sentence B it is an infinitive.

As I'm sure you realise, the two sentences are not interchangeable. A describes a situation that is common and ongoing, while B describes one that is less common and in the past.