As a native speaker of Malayalam, I don't understand Tamil aside from few words. I have been to Tamil Nadu regions especially Chennai. Tamil is very different from Malayalam although they do share some vocabularies since they belong to Dravidian Family.
Grammatically speaking, Tamil is much more closer to Kannada and Telugu than Malayalam. Malayalam is far more alienated.
Unlike other Dravidian languages including Tamil, Malayalam lacks subject-verb agreement. Here is an example below from Malayalam.
Jhan pokunnu (I am going), Nee pokunnu [You (informal) are going], Avan pokunnu (He is going), etc.
"pokunnu" is applied for all of the forms in the present tense. So verb conjugations in Malayalam is easy and also very different from other dravidian languages.
In Tamil, Kannada, and Telugu, subject verb agreement is absolutely necessary. Here is an example in the below link from Tamil.
Another difference between Malayalam and Tamil has to do with definite article.
Let me use the word for bird which is found in both Malayalam and Tamil.
Malayalam - Parava (Singular), Paravakal (Plural)
Tamil - Paravai (Singular), Paravaikal (Plural).
It must be noted that Malayalam spoken by Muslims has Arabic influences while Malayalam spoken by Christians has Aramaic (or Syriac) influences. Malayalam spoken by Hindus has more Sanskrit and Tamil influences.