Payments in lieu of notice would be if you had given your required notice that you were leaving your job, say 2 weeks notice and your employer rather than have you there for those 2 weeks working pays you the 2 weeks wages and allows you to leave your job before the notice period is over. Basically it's earned income, what you would have earned had you kept working during that notice period.
Redundancy payments are those that your employer lays you off (make you redundant) or terminates your employment and you receive a payment based on you losing your job. That is redundancy pay or severance pay. This is unearned income, what you receive as a one off payment based on either how long you'd worked for a company or good will gesture from that company, not because you'd worked for it.