Near-synonyms with slight register split
我常常去健身房 and 我经常去健身房 both mean "I often go to the gym". Native speakers use both without much thought. 经常 carries a slight connotation of regularity and routine; 常常 is a bit more emphatic on just-the-frequency. In writing, 经常 is the safer default; in speech, both appear constantly.
经常 works better in negation
Negation reveals the main practical difference. 不经常 (not often / not regularly) is natural and common. 不常常 is uncommon; native speakers usually say 不常 (dropping one character) for "not often": 我不常去 (I do not often go). For negative frequency, use 不经常 or 不常; avoid 不常常.
我不经常喝酒,偶尔跟朋友一起。
Wǒ bù jīngcháng hē jiǔ, ǒu'ěr gēn péngyǒu yìqǐ.
I do not often drink; occasionally with friends.
Position: before the verb
Both follow the subject and precede the verb (or the verb phrase including 在 for progressive): 我经常在家做饭 (I often cook at home), 他常常迟到 (he is often late). Placing them at the start of a sentence is less common and slightly emphatic. Unlike English "often" which can float, Chinese frequency adverbs prefer the pre-verb slot.
孩子们常常在公园里玩。
Háizimen chángcháng zài gōngyuán lǐ wán.
The children often play in the park.
Degree of frequency: the adverb scale
Mandarin has a frequency scale: 总是 (zǒngshì, always) > 经常/常常 (often) > 有时候 (yǒushíhou, sometimes) > 偶尔 (ǒu'ěr, occasionally) > 很少 (hěn shǎo, rarely) > 从来不 (cónglái bù, never). 经常 and 常常 sit in the "often" slot; neither leans toward "always" or "sometimes".
常 as a bare adverb
常 (cháng) by itself also means "often" and is more literary or terse: 常有 (often have), 常来 (come often). Spoken Chinese prefers 经常 or 常常; 常 alone appears in compounds and in written Chinese. For HSK and daily use, 常常 or 经常 are the production forms; 常 is recognition-only for most learners.