A clause has a subject and a verb. For example, "Bob runs track" contains the subject, "Bob", and the verb "to run". This is a clause.

So using your examples:

(1) Lemons are like Limes.

(2) He looks as if he is sick.

In the first example, you're comparing lemons to limes, a noun. In the second example, you're comparing how he currently looks to how he'd hypothetically look sick. Both underlined portions are clauses.

You may wonder why both portions are clauses (i.e. complete sentences). Technically, you could say "He looks." It's a gramatically correct sentence, but a very vague one, so a modifier is typically added (i.e. "He looks at the sky every night."). But that's another topic all together!

Continuing on...you can't say "He looks like he is sick." Again, you're comparing two clauses here, so you need to use "as".

The way I keep them straight, is to ask: Am I comparing two things directly, or what they're doing?

He runs like Bill. (Comparing he, whomever that may be, to BILL)

He runs as Bill does. (Comparing how he RUNS to how Bill RUNS).

Excellent explanation