It's really hard to explain.. 저는 영어도 가르쳐요 would be something along the lines of, (Among other subjects) I teach English too. / I also teach English. Ex. I teach science [and English too]. 저는 영어를 가르치기도 해요 would be like, (I do other things but) I also teach English / I also have a job teaching English. Ex. I'm a salesperson but [I also teach English] It's kind of hard to understand out of context, but I hope this makes more sense.. and I hope I'm right .

저는 영어도 가르쳐요 저는 영어를 가르치기도 해요 Uhm, the meaning is the same> I teach the english language. In first case the sentence is normal, But in second case, the verb 'teach' is tranformed in noun (adding 기 soon after) so it would be like "I can also do the korean teaching."