Firstly, for the overlapping meaning of "real situation", 「[実態]{じったい}」 is used far more often than 「[実際]{じっさい}」. Importantly, this is just about the only meaning 「実態」 is used for.
e.g. 「ショービジネスの実態」、「[山口組]{やまぐちぐみ}の実態」, etc. (山口組 is the largest yakuza organization.)
Very few people would use 「実際」 to say those in reality.
Secondly, 「実際」 has another meaning that 「実態」 does not, which is "actual practice (as opposed to theory)".
e.g. 「[銀行業務]{ぎんこうぎょうむ}の実際を[学]{まな}ぶ」(to study the actual practice of bank business)
You cannot use 「実態」 to say the above for the same meaning. If you do, the meaning of the phrase will change to "to study the real situation of bank business". It sounds like you are still studying with books.
Lastly, while 「実態」 is generally used only as a noun, 「実際」 is frequently used adjectivally and adverbially by adding 「の」 and 「に」, respectively.
e.g. 「実際に見てきなさい。」(Go see it in real life!) 「これが実際の[手紙]{てがみ}です。」(This is the actual letter.)