As standard Japanese is largely based off of the Tokyo dialect, there isn't all that much difference between the two. Nonetheless, there are some select expressions, words and phrases unique or overly common to the area.
"Really" (Instead of 本当 hontou or 本当に hontou ni)
"Really!?", "Is that so!?", "No way!" (as an interjection or question)
"I don't understand; I don't know" (reduction of 分からない wakaranai)
"No way!", "You're kidding me!", "Lies!" (also: うそついてる uso tsuiteru)
With verbs, it marks the progressive form, like -ing in English. (Reduction of the standard form ～ている -teiru)
食べてる tabeteru "eating"
（それは）ちょっと… (sore wa) chotto...
(That's) a little... (Used in a hesitating manner, it implies that something is a little inconvenient)
何してんの？ nani shitten no?
"What are you doing?"
"Really; super" (Adds excessive emphasis to an adjective. Avoid in standard writing.)
すごい sugoi, すげえ sugee
"Wow", "That's insane", "Amazing", "Awesome"
"I, me" (First-person pronoun, now especially common among young school girls)
Normally, -no marks an explanation and would require the copula です desu followed by the particle か ka to mark a question seeking an explanation. The latter are nearly always omitted, and の -no used on its own has come to replace the question marker か ka in most instances.
つう tsuu, てゅう tyuu
Used to mark a quoted word, utterance or statement. (Reduction of the standard form ～と言う -to iu)
なんてゅの… nan tyu(u) no... "...how do you say it... uh.."
Used to mark a quoted word, utterance or statement. (Shorter variant of the latter)
これって何？ kore tte nani? "What is this?"
"Owwwwwww!", "Ouuuuuuuch!" (Reduction and reduplication of 痛い itai "hurt; painful")
Also just いてっ ite' or いたっ ita' "ow" for short.
Imported from dialects west of Tokyo, "jan" replaces the native form じゃないか ja nai ka "isn't this...?", "isn't that so?", "right?"
これでいいじゃん？ kore de ii jan? "Isn't it enough?", "Isn't this good enough?"
～られる -rareru → ～れる -reru
The potential and passive forms that end in -rareru are now more and more frequently reduced to just -reru.
食べられる taberareru → 食べれる tabereru "able to eat", "was eaten"
Boy slang -ai, -oi > -ee
The sequences /ai/ and /oi/ can be replaced by /ee/ for aesthetic purposes, giving the allusion of a more marked "manly" slang, or "rough" speech. So a word like すごい sugoi would become すげえ sugee, and 赤い akai "red" would become あけえ akee. This feature is usually more prominent in Western dialects, while in Tokyo it's just an imitative device. So it's not usual for someone to completely replace all ai's and oi's by ee's, but it is common to do so with certain more slang-like adjectives like sugee.
Devoicing and voicing
Devoicing is especially marked by men, where words like いち ichi "one" and 好き suki "like" are essentially pronounced like いっ "ih" and "s'ki". The vowels are a little more noted among women, but may be fully pronounced to add a certain level of aesthetic, formality or pompousness, especially with verbs and endings like です desú and ～ます -masú (accent used to mark displaced stress).