I agree with @aper's answer, and would like to add some more color.
To get a general sense and feeling of the difference between animus and anima, one can consult my Collocation Tool to see what words are usually paired with animus and anima.
under animus we can see words like aequus, moveo, magnitudo, magnus, bonus, fortis, induco, pertubo, rubor, dolor, cogitatio and many more.
under anima we find: caro, vivo, humanus, amitto, immortalitas, mors. (those words are not to be find that high in aminus).
From that survey one can, I believe, start getting sense of the difference between the terms. anima seems to be much more oriented towards the very (base) function of life. hence amitto/depono/efflo anima with meaning "to die". While animus tends to relate to a higher function of the human life like attention and character, feeling, will, etc.; hence aequo animo (calmly), or ab perturbatione animi liberatus (free from any passion/disturbance of mind). Actually anima can be sometimes used to designate life itself and replaceable with vita:
“Difficile est,” inquis, “animum perducere ad contemptionem animae. (Seneca) [It is difficult, you say, to move the aminum towards a contempt of animae (=life)]
For a more closer and accurate analysis of the differences, one can consult some synonyms dictionaries, one of those which is also in the public domain and available in readable format is of Döderlein’s. Thus he writes:
Anima denotes ‘the soul,’ physiologically, as the principle of animal life, in men and brutes, that ceases with the breath, like ψυχή: animus (ἄνεμος), psychologically and ethically, as the principle of moral personality, that ceases with the will, like θυμός. The souls of the departed also are called, in a mythological point of view, animæ, as shades; but, in a metaphysical point of view, animi, as spirits. Anima is a part of bodily existence; animus, in direct opposition to the body...
Another one, for the Latin readers, is De differentiis verborum by Popma, which is similarly available today in a readable edition. After establishing the kind of difference we just mentioned (mainly from examples and other previous sources who already dealt with that question), Popma thus remarks:
... Cicero tamen et Seneca saepe dicunt animi immortales, et immortalitas animorum. Hanc differentiam tamen non observat Sall. B. C. I. Immo ipse Cic. frequentissime animum ponit pro anima, v. g. Tusc. Disp. I. 31. Somn. Scip. IV. 3. et 8. ...
Popma basically warns us that this difference is not so set in stone, and should be taken with a grain of salt. He notes that Cicero and Seneca used the expressions animi immortales and immortalitas animorum(which we would naturally expect to have with anima not animus), and says that Cicero frequently used animus "instead" of anima.
More to remember, as @Rafael notes in his comment to other answer, the reader should bear in his mind that the nuances tend to change across time and space.