a_problem.js

var Promise = require("bluebird");

function doSomething()
{
    return Promise
        .try(function ()
        {
            return task1();
        })
        .then(function (result)
        {
            return task2(result);
        })
        .catch(function (error) //Best way to only catch task2?
        {
            return processError(error);
        })
        .then(function (result) //Should not run if task1 throws...
        {
            return keepOnGoing(result);
        });
}

possible_solution1.js

var Promise = require("bluebird");

function doSomething()
{
    return Promise
        .try(function ()
        {
            return task1();
        })
        .then(function (result)
        {
            return Promise
                .try(function ()
                {
                    return task2(result);
                })
                .catch(function (error)
                {
                    return processError(error);
                })
        })
        .then(function (result)
        {
            return keepOnGoing(result);
        });
}

possible_solution2.js

var Promise = require("bluebird");

function doSomething()
{
    return Promise
        .try(function ()
        {
            return task1();
        })
        .catch(function (error)
        {
            error._some_unique_identfier = true;
        })
        .then(function (result)
        {
            return task2(result);
        })
        .catch(function (error)
        {
            if (error._some_unique_identfier) 
            {
                throw error;
            }
            else 
            {
                return processError(error);
            }
        })
        .then(function (result)
        {
            return keepOnGoing(result);
        });
}