this sentence ends with です
田中さんも有名です。
while this one does not
服も古い。
please help me clarify the nuances.
thank you!
The first one is more formal than the second one
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There’s a difference in formality. The first sentence is using a more formal copula. The second sentence would be using a less formal copula, if it were using a copula at all 
A copula is basically a word to tie together a subject, and a predicate describing the subject. In many western languages, the verb meaning “to be”, “to exist” is kind of re-used to function as the copula:
The question is simple.
Here, “the question” is the subject, and the adjective “simple” describes the subject. Both are tied together by the copula, which is “is”, a form of “to be”.
In Japanese, there are multiple level of formality that can be hard to express in many western languages. You could translate the example above as
問題は簡単です。
or as
問題は簡単だ。
The first version uses the more formal, more polite copula 「です」, while the second version uses the less formal, more direct copula 「だ」.
Therefore, the first version is more formal, more polite. You might use it when talking to a stranger or a person you want to show respect to, like a teacher. The second version is a bit less formal, more direct. You might use it when talking to somebody you know well, like a close friend.
But why doesn’t your second example,
服も古い。
show any copula at all? Because of the adjective.
As you may know, there are two types of adjectives in Japanese that behave differently. In the example I’ve shown above, I’ve used the adjective 「簡単」, which is a so called “na adjective”. You can use both copulas we’ve seen, 「です」 and 「だ」, with na adjectives.
But 「古い」 is a so called “i adjective”. You can use 「です」 with i adjectives, but not 「だ」. Therefore, you could say
服も古いです。
but you couldn’t say
服も古いだ。
This needs to be, as in your example,
服も古い。
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This is a bit beside the question, since it wouldn’t be applicable almost ever, so sorry about that. But the い-Adjectives grammar point also has this as its “Fun Fact”:
Although using です with い-Adjectives is very common, it is actually incorrect Japanese. As です is simply a polite variation of だ, it follows all of the same conjugation rules as だ. This means that it should only be attached to nouns and な-Adjectives. However, as most Japanese natives do not know this rule, it is widely considered natural to use.
Which I guess could mean that in some places that are very strict (to the level that most native speakers aren’t aware of) about correct form, です wouldn’t be used with い-adjectives either?
However, some cursory googling reveals people claiming that using です with い-adjectives would’ve been accepted as correct in standard Japanese in 1952, so I’m not sure if the Fun Fact is accurate in this case? For example, in this article (‘A study on the spread of the usage of an auxiliary verb “desu”’, 2018):
1952 年 4 月，国語審議会は「これからの敬語」(１)（７ 形容詞と「です」）で，「これまで久しく問 題となっていた形容詞の結び方――たとえば，『大きいです』『小さいです』などは，平明・簡素な形 として認めてよい。」と建議し，学校教育もその方針に沿って文法教育がなされてきた
This seems to be text of the publication mentioned above, containing what the article claims.
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This fun-fact really depends on which ‘expert’ you ask, as many Japanese linguists are on the fence about it. The bottom line is, almost nobody (except maybe a very strict professor) will ever claim or say い-Adj+です is wrong.
We mainly chose to include that fun-fact just to show that there isn’t some magical difference between だ, and です, apart from the accepted norm changing over time.
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