For the headline question, no. See, for example, Wilkinson's polynomial: http://en.wikipedia.org/wiki/Wilkinson's_polynomial.

Edit: From the response to this answer, it seems that some expansion is needed. The OP asks two questions. In the the title, and repeated in the text, the question is: Do approximately the same polynomials have approximately the same roots? The answer to this question is no, as Wilkinson shows with his monic degree-20 polynomial, which has well separated roots of moderate size, namely the integers 1 to 20. A decrement of $2^{-23}$ in the coefficient of the degree-19 term (which is $-210$)---a proportional change of less than 1 in 1.7 billion---induces substantial changes in the roots: for example, the root pair $16.5\pm 0.5$ becomes the root pair $16.73024\pm 2.8162\mathrm i$ (to 5 d.p.). The second question asked by the OP is whether the coefficients-to-roots map is continuous, which was already answered (affirmatively) by others.

For the headline question, no. See, for example, Wilkinson's polynomial: http://en.wikipedia.org/wiki/Wilkinson's_polynomial.

Edit: From the response to this answer, it seems that some expansion is needed. The OP asks two questions. In the the title, and repeated in the text, the question is: Do approximately the same polynomials have approximately the same roots? The answer to this question is no, as Wilkinson shows with his monic degree-20 polynomial, which has well separated roots of moderate size, namely the integers 1 to 20. A decrement of $2^{-23}$ in the coefficient of the degree-19 term (which is $-210$)---a proportional change of less than 1 in 1.7 billion---induces substantial changes in the roots: for example, the root pair $16.5\pm 0.5$ becomes the root pair $16.73024\pm 2.81262\mathrm i$ (to 5 d.p.). The second question asked by the OP is whether the coefficients-to-roots map is continuous, which was already answered (affirmatively) by others.

For the headline question, no. See, for example, Wilkinson's polynomial: http://en.wikipedia.org/wiki/Wilkinson's_polynomial

For the headline question, no. See, for example, Wilkinson's polynomial: http://en.wikipedia.org/wiki/Wilkinson's_polynomial.

Edit: From the response to this answer, it seems that some expansion is needed. The OP asks two questions. In the the title, and repeated in the text, the question is: Do approximately the same polynomials have approximately the same roots? The answer to this question is no, as Wilkinson shows with his monic degree-20 polynomial, which has well separated roots of moderate size, namely the integers 1 to 20. A decrement of $2^{-23}$ in the coefficient of the degree-19 term (which is $-210$)---a proportional change of less than 1 in 1.7 billion---induces substantial changes in the roots: for example, the root pair $16.5\pm 0.5$ becomes the root pair $16.73024\pm 2.8162\mathrm i$ (to 5 d.p.). The second question asked by the OP is whether the coefficients-to-roots map is continuous, which was already answered (affirmatively) by others.