While she was enjoying her coffee she suddenly noticed a hand from behind SNATCH her handbag. This sentence is correct. The past tense (noticed) interrupted the past continuous tense (was enjoying). Maybe you'll understand these examples: During the hurricane, we heard a tree fall. While we were eating, we watched the performers dance.

Since there are 2 actions happening at the same time (enjoying her coffee, and the thief snatching her handbag), the second action is in the infinitive. Using -ing at the end of the first verb establishes that it is a current action, so it would be redundant to use -ing at the end of the second verb.

Honestly, English is just one of thoes languages that don't make much since. You could also say snatching or snatched but just saying snatch feels more natural in this sentence.

Have a look at https://linguapress.com/grammar/infinitive.htm It includes : ("noticed" qualifies here as a "verb of primary perception") After verbs of perception, the second verb can alternatively be a present participle: I heard him leave the house by the front door. or I heard him leaving the house by the front door. Both these structures are possible, though there may be a shade of difference between the two; normally the speaker can choose. Alternatively look at https://dictionary.cambridge.org/grammar/british-grammar/verb-patterns/hear-see-etc-object-infinitive-or-ing 1.1. The short infinitive , without to This is used notably with certain modal auxiliaries, can, could, may, might, will, shall, could, must. It is also found after a handful of other verbs that introduce a verb complement, in particular: dare, verbs of primary perception see, hear, smell, feel, and some verbs of permission or causative verbs, notably make, let and have. Finally there are two common words that are followed by the infinitive without to: these are rather and better, in expressions on the model I'd rather.... After verbs of perception, the second verb can alternatively be a present participle: I heard him leave the house by the front door. or I heard him leaving the house by the front door. Both these structures are possible, though there may be a shade of difference between the two; normally the speaker can choose.