a person is considered as an partner by estoppel when he/she unkowningly acts or behaves in such a way that the third party considers them to be a partner in the firm. for eg. Karan is rahul's good friend who is a partner in Apex.pvt ltd. One day Karan accompanies Rahul to one of the firm meetings with a third party and pays a major role in the negotiations. now, the tnird party considers him to be a partner in the firm.In the future if the negotiations with two party breaks down Karan will be held liable for any loss incurred.
Whereas, in partner by holding out a person with his consent allows knowingly the use of his/her name as the partner of the firm even if they are not. In such a case any liablity which is made because of this representation, the person will have equal liability.