Can спешить replace торопиться in this sentence, for example:
“на вашем месте я не торопился бы критиковать все направо и налево…”
Спешить and торопиться are synonyms and there's no real difference in their meaning. I'm struggling to come up with an example where it would be appropriate to use one word but not the other.
Specifically, in the sentence you provided, you could substitute спешить for торопиться without any change in the meaning, not even the slightest:
“на вашем месте я не торопился бы критиковать все направо и налево…”
“на вашем месте я не спешил бы критиковать все направо и налево…”
Edit: you can see in the synonyms dictionary that торопиться is the first word listed for спешить: http://словарь-синонимов.рф/words/спешить
Edit 2: @DmitryAlexandrov brought up a very good example where these words are not interchangeable, but generally they totally are.
In the general case the only pattern, that came into my mind, when you can not interchange спеши́ть and торопи́ться is a talking about a gaining clock:
Часы спешат на пять минут [The watch is five minutes fast]
but extremly rare часы торопятся.
часы торопятся
. Especially among older people.
There is some difference in imperative mood.
Торопись!
Means drive, urge on yourself.
Спеши!
Not really used. If used, I would understand it as "be hurry in panic". I hardly can imagine where such meaning would be appropriate.
In plural it may be used in advertising of a performance
Спешите посмотреть новое представление!
Be hurry in panic to see the new performance because it is very interesting (one can instantly imagine crowds storming the circus or the theater, which is possibly the organizers agitate for). This usage is somewhat outdated I think, at least I encountered it only in books.
Торопитесь посмотреть новое представление!
This just means that you can be too late to see the performance because it will be ceased being shown in a few days. You are not agitated to run to the theater, losing your bags and spectacles on the way, you just reminded so that you possibly better to review your plans accordingly and postpone other deals.
Just a side note - there is a saying: "Торопись, не спеша", - which means "Hurry up without hustle" - so try to do things faster, but do not mess them up by doing so.
To my mind only idiomatic use can show any difference, for instance, you can say "поспешишь, людей насмешишь" - and there is no other way to put it.
They do mean the same, but watch for the indirect meaning, indicating that a person does things with insufficient consideration. Sometimes, the only reason why торопиться and спешить seem so different is that implied indirect meaning of these words.
Спешащий and торопящийся have the same meaning, but they are not the same as спешный and торопливый.
Tоропиться and спешить -- is not the same as поторопиться and поспешить. "Поспешить -- людей насмешить"
Спешить is a description of the action.
Я спешу к поезду. Мне некогда с тобой разговаривать.
Я спешу на встречу.
Я спешу сообщить тебе что-то.
Торопиться is a description of the circumstance.
Я тороплюсь на поезд, поэтому у нас есть лишь 5 минут.
Я слишком тороплюсь! (и потому рискую ошибиться).
Я тороплюсь как могу/изо всех сил.
Not in a direct context of the question, but related to it giving some additional, not presented yet, info. Because also the word спешить
has another Semantic property or meaning property in despite the writing/spelling is the same:
Спешить - вынудить спешиться, то есть сойти или упасть с коня, велосипеда и т. п., выйти из транспортного средства —
"Какой-то французской офицер объезжает передовую цепь. — Не спешить ли его? — шепнул второй часовой, взводя курок."
Opposite semantics, or antonym:
посадить, усадить (например, в седло)
I've grown to liken "торопиться" to "to hurry" and "спешить" to "to rush".
If you can think of English expressions where one of them is the preferred, or indeed the only choice, we could do an interesting experiment by trying to translate them to Russian.
торопиться
is a little bit bookish. Very little bit :)