There are two expressions below:
1. A drowning man will catch a straw.
2. A drowning man will catch at a straw.
Please, tell me the difference between the two sentences above in meaning.
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I think "grasping at straws" is the most common way to phrase this English idiom. It means you are getting so desperate that you reach for solutions which are very unlikely to actually work (or you are just making guesses without any real idea of whether it could work at all – similar to "a shot in the dark").
The Wiktionary entry for "grasping at straws" also refers to a drowning man, so I suspect wherever you encountered "catch a straw" it was just a mis-translation of the more common "grasp at straws."
As others noted, even if you ignore the idiom, the phrases using "catch" are both unusual wording. You wouldn't "catch a straw" unless it was running away from you or needed to be trapped (e.g. "catch a butterfly"). For general usage (ignoring the idiom), "clutch at" is more correct than "catch at."