The simple present tense is expressed with 終止形 of verbs and aux verbs, which is the same with the base form.
- 私はアメリカ人です -> I am American
- 気分はどうですか？ -> How is your feel? How are you feeling?
- 彼は野球の選手だ -> He is a baseball player
- 僕もそう思う -> I think so too
- あなたはどう思いますか？-> What do you think?
- ここを押すと痛みを感じますか？ -> Do you feel any pain when I push here?
The continuous/progressive aspects are formed as 連用形 of verb+て(で)+いる
- 彼女は寝ている -> She is sleeping
- 彼は本を読んでいる -> He is reading a book
The continuous/progressive aspects are often followed by ところ, 最中, 真っ最中, 真只中, etc.:
- 彼は今エンジンオイルを交換しているところだ -> He is in the middle of changing engine oil
- 風呂に入っている最中に宅配便が来た -> There was a delivery when I was taking a bath
- 今試験をしている真っ最中なので部屋には入れません -> You cannot enter the room because we are right in the middle of an exam now
What is expressed in the simple present in English is often expressed using the continuous/progressive in Japanese:
- 彼女は大学で数学を教えている (❌教える) -> She teaches math at college
- 彼はパン屋を経営している (❌経営する) -> He owns and runs a bakery
- 私は車を二台持っている (❌持つ) -> I have two cars
- 私達は新製品に自信を持っている (❌持つ) -> We have confidence in our new product
- 私は夫を愛している (❌愛する) -> I love my husband
Japanese tend to use the continuous aspect for the imperfect aspect:
- 彼、私のことをどう思っているのかしら？ (❌思う) -> I wonder what he thinks of me
- 彼は私のことを良く思っていない (❌思う) -> He doesn’t think highly of me
- 彼は自分が天才だと思っている (❌思う) -> He thinks he is a genius
- 彼は犯罪に関わっている (❌関わる) -> He is involved with criminal activities
- 彼の事はよく覚えていない (❌覚える) -> I don’t remember him well
- その団体は政府の補助を受けている (❌受ける) -> The organization is receiving the government subsidy
- いけないと分かっていても止められない -> I know I shouldn’t but I can’t stop
However it is not a set-in-stone rule, and there are many other cases:
- 僕は彼女は美人だと思います (❌思っています) -> I think she is beautiful
- 彼の話はどうも怪しいと感じる (❌感じている) -> I feel his story is really fishy
- 私は彼とは合わない (❌合っていない) -> I don’t see eye to eye with him. I don’t like him
- 酒は日本酒に限る (❌限っている) -> I limit alcohol to sake. I think sake is the best
In other cases either is fine:
- 彼は娘をとても可愛がる/可愛がっている -> He adores his daughter so much
- 彼の話は辻褄が合わない/合っていない -> His story doesn’t add up
- 彼等の製品はまだ競争相手にまだ劣る/劣っている -> Their products are still inferior to their rivals
- 私は彼を信用します/信用しています -> I trust him
And in yet other cases the simple present and the continuous/progressive mean two different things:
- 彼は話が分からない -> He is not capable of understanding. He is unreasonable, pigheaded
- 彼は話が分かっていない -> He is not understanding the circumstances, what I told him
- 僕は車を持たない -> I don’t and won’t have a car (because I think cars are not good, necessary, etc.)
- 僕は車を持っていない -> I don’t have a car (but I will if I can afford it, if I really need it, etc.)
In many cases, it is purely idiomatic. The below two are the same thing, just using different phrasings:
- 彼は危ないと知っていながら何もしない -> He doesn’t do anything even though he knows it is dangerous
- 彼は危ないと知りつつ何もしない -> He doesn’t do anything while he knows it is dangerous
Many aspects of Japanese language are highly idiomatic and cannot be clearly defined by rules. One thing that can be mentioned is, aside from aux verbs だ/です and ます, the simple present sounds slightly literary, while continuous/progressive sound more conversational.